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-1926690.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 148 浏览量
2024-03-17
21:35:42
上传
评论
收藏 5.74MB PDF 举报
温馨提示
试读
31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
1/26/2019 MCM 2019 Summary Sheet for Team 1926690
https://www.comap.com/undergraduate/contests/mcm/team-summary-sheet.php 1/1
Print This Page
Close This Window
For office use only
T1 ________________
T2 ________________
T3 ________________
T4 ________________
Team Control Number
1926690
Problem Chosen
A
For office use only
F1 ________________
F2 ________________
F3 ________________
F4 ________________
2019
MCM/ICM
Summary Sheet
(Your team's summary should be included as the first page of your electronic submission.)
Type a summary of your results on this page. Do not include
the name of your school, advisor, or team members on this page.
Differential Equations and Dragons: Gold Nuggets of Wisdom
From The World’s Most Terrifying Role-Playing Game
Team 1926690
January 29, 2019
Abstract
A truly terrifying future may await inhabitants of the earth: the release of three full-grown dragons
into the world. In this analysis, we examine the ecological impacts of dragon introduction, in particular
for the purposes of informing fictional literature and advancing real-world population assessments.
First, we present a global agent-based model which we designed to understand the global distri-
bution of dragons and their environmental impact, as well as to evaluate the potential result of human
intervention strategies. Three dragons are initialized in the object-oriented model and given individual-
level characteristics that contribute to their decision-making and the ecological havoc they eventually
wreak.
Next, we develop a regional differential equation model that zooms in on a single dragon and
investigates the dynamics of this dragon’s growth and its impact on and the population of prey species
and the vegetation cover in its range. Examining a wide range of parameter sets, we find that our dragons
are voracious eaters that will likely prove to be a nuisance for human populations. In response, we posit
that a likely outcome of the dragon introduction event is management of the creatures by housing them
in zoos, and we calculate the food resources necessary to support dragons held in captivity. Given that
the caloric requirements of our modeled dragons require them to consume thousands of deer each year,
we recommend that the dragons be held to a strict diet to limit their growth, in which case the dragons
still require hundreds of deer at their physiological minima.
Our models indicate that releasing dragons into the wild could have disastrous ecological impacts,
including the elimination of forests, melting of sea ice, and decimation of many prey populations. How-
ever, since dragons may provide benefits, such as regulation of nuisance populations and provenance
of resources to humans, careful management could turn the ecology of the situation around. Because
of the volatile nature of this problem, we devised multiple complementary models to best capture the
problem and its potential solution. This approach is of value to population managers dealing with similar
environmental concerns with real-world nuisance species and human-caused habitat destruction.
Team 1926690 Letter to George R.R. Martin
Team 1926690
Dragon Den Modeling Agency
George R.R. Martin
Super Secret Medieval Lair
Santa Fe, NM
Dear Mr. Martin:
Our team has conducted an analysis regarding the likely outcome if the three fictional dragons from Game
of Thrones were to be released into the world, encompassing their prey requirements, land use strategy,
and reproductive success. We based our modeled dragons on your story A Song of Ice and Fire, with some
notable modifications to accommodate the likely needs and preferences of fictitious dragons released into the
real world. We wish to present some recommendations for how you might incorporate this fictional element
of your book meeting the real world into your next story.
To begin, we devised an agent-based model which randomly placed the three dragons, with semi-randomly
assigned characteristics, somewhere on the map, which we represented as a series of 45 connected regions.
While we did not initially place limitations on the areas in which the dragons could live, it quickly became
apparent that these dragons would gravitate toward semi-arid temperature or tropical regions, given an
assumed preference of these cold-blooded creatures for warmer climates and ample land availability for
nesting. Although we did not encode the volcanic activity of any particular region, we did assign the 45 map
locations a number of caverns in which the dragons could nest. Given the requirement of mature dragons in
our model to locate gold nuggets to create a permanent den, which we derived from detailed dragon analysis,
natural resource availability quickly emerged as an important constraint on dragon growth and population
expansion.
Through our agent-based model, we found that the habitat range of dragons tended to be relatively limited
and that dragon populations, if left unchecked, tend to completely rob a region of its natural resources. For
this reason, if you choose to write a story on the real-world implications of dragon escape, it is likely that
you will either need to imagine a post-apocalyptic world, or will need to design a mechanism through which
humans can manage the dragon population. In our model, we included the potential intervention strategy
of people removing a maximum of one dragon from each region each day in response to habitat destruction
imposed by the dragons. People were also able to restore some of the removed forest habitat in our model.
We found that the most likely regions for dragon dominance were tropical or sub-tropical regions of the
globe where there is a lot of land (thus many available caverns), including the Americas (in particular, the
continental United States and northern South America) and some parts of Asia and Africa.
We find that dragons are unlikely to be able to live for long periods of time in the arctic regions due to their
requirement for prey species. Dragons are, however, likely to find a home in wetter, more temperate climates
as opposed to more arid regions. Dragons that prefer to live in the arctic would probably need to travel long
distances to accommodate their appetites, especially given potential forthcoming population declines in the
arctic.
In response to the constraints we found on global resource availability, we explored both the baseline ecological
impacts of these dragons and the potential interactions humans might have with the newest member of the
animal kingdom. To gain a better understanding of the impact a single dragon might have on a region, we
designed a differential equation model to simulate the lifespan of a dragon and its impact on prey species
Team 1926690 Letter to George R.R. Martin
(assumed to be deer) and local vegetation. We found that, in order to grow to full size, a dragon must
be able to meet its nutritional needs, meaning that there must be an ample supply of deer available and
the dragon must be somewhat good at hunting. Specifically, there must be a sizeable herd of deer initially
available, and these deer must multiply at at least a rate of 15% per year in order for the dragon to have
enough food to eventually reach at least 83% of its target weight. We computed the caloric demands of a
dragon using a power law and found that a fully grown 6,000 kg dragon (the mass of an elephant) needs to
consume at minimum 2,647 deer per year to sustain its metabolism. As a high estimate for deer population
density would be 45 per square mile, a dragon would need at least 58 square miles of land to have enough
deer for a year, and then it wouldn’t have any left for the next year. Although we are supportive of the
flourishing of dragons, unfortunately, in most scenarios we explored, these ruinous reptiles decimated the
deer and evicted the vegetation in the areas they inhabited.
Since the energy requirements for a dragon are so great and ecological impacts so dire, we find it probable
that these dragons would not be allowed to run amok in the wild, but would rather be kept in captivity.
This could be an interesting plot element in your story that is a realistic outcome, but it requires that you
set your story in modern times. We propose that dragons, either before or after initially wreaking havoc
on global ecosystems, could be kept in zoos. We conducted a detailed analysis of the resources that must
be furnished to a captive dragon, and have found that a dragon kept in captivity restricted to only 250 kg
in eventual size would require 158 deer per year to sustain, or 2,627 deer to approximate what the dragon
would likely eat in the wild. This would warrant a tremendous investment in the care of each animal, which
would be possible for only the world’s largest and most capable zoos, and would also present an animal rights
quandary if the dragons were only provided with the bare minimum. The spectacle of an in-house dragon
would be a significant attractor for visitors to said zoo, likely visiting from locations around the globe. This
high concentration of people would be perfect for a plot twist involving some dramatic action terrorizing the
crowd.
We also posit that dragons could become part of human culture by serving as a food source, in addition to
their ecological role in managing extremely large prey populations. This would be a particularly interesting
plot element in a story set in medieval times. We find through our modeling that dragon populations have
the potential to grow substantially in number and would likely require management, in particular through
hunting. Humans could also use dragons for transportation, increasing the likelihood that they would
overlook their role in habitat destruction, thus improving the probability that the dragons could peacefully
cohabit with people.
After completing this ecological assessment of the potential woes of global dragon infestation, we are confident
that, although the situation may mean environmental calamity, it is rich with content for your potential use
as a writer. We welcome additional inquiries about the ecological dimensions of a dragon release event.
Most sincerely,
Team 1926690 of the Dragon Den Modeling Agency
Team 1926690 Page 1 of 20
1. Introduction Dragons have long captivated the human psyche. From the ancient Near East and
Mesopotamia to the High Middle Ages of western culture to the present, dragons repeatedly appear as
large, reptilian creatures in folklore and mythology [1]. Modern audiences may imagine dragons as sinister
hoarders of gold and jewels, as depicted in Tolkien’s The Hobbit [2], magnificent and perhaps misunder-
stood in J.K. Rowling’s Harry Potter series [3], and even alluring suitors of donkeys in Dreamworks’ Shrek
franchise [4]. While conceptions of dragons have varied with geography and time, the typical dragon in
western culture today is a large, winged, fire-breathing reptile of enormous strength and often capricious
temperament, with build ranging from serpentine to Tyrannosaurus rex -esque.
In the modern hit television series Game of Thrones, based on George R. R. Martin’s fantasy series A Song
of Ice and Fire, three dragons are raised by Daeneryus Targaryen, the “Mother of Dragons.” While the
setting of Game of Thrones is fictional, its dragons raise the question of whether it could be sustainable to
raise dragons on the earth. Dragons from the series tend to favor volcanic mountains as habitats, but as the
series has begun to suggest, it is also possible these creatures might thrive in human-dominated habitats.
2. Problem Statement In this report, we analyze the feasibility of three dragons, based loosely on those
in Game of Thrones, living on the earth. We analyze dragon characteristics, behavior, habits, diet, and
environment and assess the dragons’ ecological impact and requirements, energy intake and expenditures,
and land management requirements. We examine these questions in biomes with nine possible distinct
temperature and dryness regimes, and we analyze both the impact of dragon migration between biomes and
the influence of single-dragon introduction in a particular habitat range. For modeling purposes, we assume
that the dragons are large reptiles that can fly long distances, breathe fire, and resist attack by any other
species. The only assumptions we consistently maintain about the dragon’s growth and development is that
dragons are born weighing 10 kg and grow to 30-40 kg within a year.
3. Data Sources and Modeling Approach To better inform our models, we assimilated some physio-
logical data for comparable organisms from the scientific literature. We also set strict guidelines for dragon
growth, reproduction, and feeding which we conserved across our different modeling approaches. We thus
cohesively investigate dragon life history, which could be important to understand dragon ecology.
We first implemented a macroscale, agent-based model for dragon expansion across the globe, incorporating
constraints on permanent settlement formation (“den” location), energetic requirements, habitat destruction
by dragons, migration, and reproduction. We then implemented a microscale differential equation model of
the growth and development of one dragon in a fixed region and its ecological impacts.
Upon reviewing the results of the agent-based and regional differential equation models and realizing that
allowing dragons to roam freely in the wild would be highly detrimental to the environment, we implemented a
differential equation model for dragons in captivity to gain better insight into the support they would require.
4. Modeling Methodology
4.1. Unique Model Characteristics For the purposes of the agent-based model, we split the globe up
into regions of a size which we deemed reasonable for daily dragon movement. Thus, the maximum distance
a dragon can travel per day is the space of an “ecoregion” cell in model space. Each region was assigned
traits which would help or hinder dragon development and reproduction as well as impact its environment.
Gold Nugget Nesting and Reproduction In a nod to Smaug from Tolkien’s The Hobbit [2], we decided
that a unique feature of our dragons would be that they require gold in order to make their nests. To
facilitate this, we mandated that each dragon would need to mine for gold resources according to known
gold reserves in the territories it inhabits. Once a dragon garners enough gold to produce a suitable den, it
settles down and is able to reproduce either sexually or asexually. Asexual reproduction comes at a greater
energetic cost, thus is only chosen as a strategy when the dragon is sufficiently healthy and no mates are
available. For the purposes of simplification, we assume that all interactions between adult (age ≥ 1 year)
dragons can result in sexual reproduction, and do not make a distinction between male and female dragons.
Team 1926690 Page 2 of 20
Figure 1: Our partitioning of the world into 45 ecoregions, coded above by their dryness, with blue shading
representing a wet climate, orange a semi-arid climate, and yellow an arid climate. Credit for underlying
map: Wikimedia Commons.
This decision to give dragons multiple reproductive strategies follows from biological phenomena observed
in similar reptiles. In the Komodo dragon, reproduction may occur via either asexual reproduction through
parthenogenesis (virgin birth) or sexual reproduction [5]. We assign a more substantial health cost to asexual
reproduction in our model, but do not enforce that dragons substantially alter their behavior (i.e. migrate)
in order to find mates, particularly since we assume that a dragon must settle in a den before it is able to
reproduce.
In the regional differential equation model, neither reproduction nor gold-mining behaviors are captured, as
the primary focus of that model is to understand the ecological impacts and demands of an isolated dragon.
Dragon Prey We were not specific about the type of dragon prey in our global agent-based model; when
we initialize prey species, we give them random caloric contents. In the regional differential equation model,
we assume that the dragon, being at the top of the food chain, may eat a variety of prey animals; however,
we refer to all prey animals as “deer” for succinctness and as a nod to the fact that we base the caloric content
of the prey animals off that of a deer [6], one of the most widespread large land animals in the world.
4.2. Agent-based Modeling
Motivation To model worldwide population dynamics of dragons after initial release of three “mature”
dragons into random global ecoregions, we opted for an agent-based model (ABM). These models are often
used for population studies requiring the analysis of multiple individual-level characteristics [7], which may be
stochastic in nature. We used our ABM, implemented as an object-oriented model in Python, to understand
the extent of potential dragon population spread, both in terms of geography and sheer numbers. The results
of this investigation inform our later modeling and data analysis approaches for the purposes of stress-testing
different environments under dragon pressure, and understand whether intervention measures are necessary.
Mapping To facilitate movement of dragons across different areas of the globe, we divided the world into
45 distinct ecoregions, each of which we assume is sized such that our dragons could move from one to the
other within the course of a day. We assigned several characteristics to each region, representing factors that
would impact healthy growth and reproduction for each of the dragons.
Earth and Region Initialization In the agent-based model, first an “Earth” object is created, into which
each of the forty-five regions are placed. The three dragons are then randomly placed on the map, with the
剩余30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