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2019美赛O奖论文-MCM2019A-1902727.pdf
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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
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For oce use only
T1
T2
T3
T4
Team Control Number
1902727
Problem Chosen
A
For oce use only
F1
F2
F3
F4
2019
MCM/ICM
Summary Sheet
Dragon and Ecology
Summary
This paper analyzed dragon’s biological attributes, energy requirements and its interraction
with environment by stepwise modeling.
To study the relationship between dragon’s weight w and its age a, we apply Logistics model
after biological analysis. Then we use provided body mass data of dragon at dierent age, to t
the weight-age curve. It was found that the dragon grew rapidly from age 3 to age 5, then almost
reached the maximum at age 8, and then grew slowly. The maximum body weight was about
14.8 × 10
5
kg.
Then we set a model to nd the relationship between daily energy consumption q, daily caroles
intake u and age a of the dragon, according to Kleiber’s law (the relationship between the energy
consumption and body weight w is q ∝ w
3/4
) and the law of biological growth(consumption and
expenditure determines the weight increase). An adult dragon would eat around 4.8×10
6
kcal per
day.
Under dierent climate conditions, area required to support a dragon varies. To simplify,
we refer to the novel and we found dragons mostly eat sheep. So the food chain is herbage,
sheep, dragon. First we calculated how many sheep will 3 dragons eat per day, then we use
Malsus model to estimate the required sheep population. To analyze how mush area is required
to support sheep group, we set a Integral regression model to calculate herbage yield, and use
realworld temperature, rainfall, sunshine data to quantize it. In arid, temperate, arctic area,
about 3738hm
2
, 2383hm
2
, 4843hm
2
of pasture is required respectively.
Providing dierent levels of assistance to dragons corresponds to various community size, we
dened 7 levels corresponding to 7 types of ecosystem. Using unit area biomass of them, we get
the community size. When it’s on level 5, dragon lives in temperate grassland, 3.81 × 10
7
kg of
biomass is required.
An active dragon will aect surrounding ecosystem, we use species extinction speed, NPP(net
primary production) and biomass decrease to measure degradation of ecosystem. E.g. a grown-
up dragon will wipe out a 20, 000 group of cattle in 103 days, also let a temperate evergreen
forest degrade to grassland, so the NPP will decrease 1.75 × 10
7
per year, biomass will decrease
8.35 × 10
8
per year.
In practical, the model can be applied to analyze agribusiness problems, or invasive species.
This paper also analyzed the model’s sensitivity to dragon’s origin weight data , if it’s increased
by 3%, the calculated adult weight will change by 5.36%, its daily energy expenditure will change
by 8.98%, so the origin data has aect on model’s stability, but the result is still acceptable.
Keywords: Logicstics, Intake and Expenditure, Ecosystem Balance, Integral Regression
Team # 1902727 Page 1 of 23
Contents
1 Overview 2
1.1 Backgroud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Restatement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Assumptions and Justications 2
3 Notation 4
4 Analysis and Modeling 4
4.1 Dragon’s weight growth equation based on Logistics Model . . . . . . . . 4
4.2 Energy consumption and intake model of dragon based on Kleiber’s law . 6
4.2.1 Energy consumption . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.2.2 Energy intake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.2.3 Comparison of energy consumption and intake . . . . . . . . . . . 7
4.3 Predation area required to support dragon under dierent climate conditions 8
4.3.1 Flock size under ecological equilibrium . . . . . . . . . . . . . . . . 9
4.3.2 Herbage yield under dierent climate conditions . . . . . . . . . . 10
4.3.3 Predation area needed to support dragons in three types of regions 12
4.4 Community required to provide dierent level of assistance for dragons . . 13
4.5 Impact of dragons on the ecological environment . . . . . . . . . . . . . . 15
4.5.1 Extinction rate of species . . . . . . . . . . . . . . . . . . . . . . . 15
4.5.2 Degree of degradation in production capacity and declination in
biomass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5 Sensitivity Analysis 17
5.1 The sensitivity analysis of dragon’s weight . . . . . . . . . . . . . . . . . . 17
5.2 Reliability verication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6 Strengths and Weaknesses 18
7 Practical Application 19
7.1 In agribusiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
7.2 Predicting the eects of invasive species . . . . . . . . . . . . . . . . . . . 20
Team # 1902727 Page 2 of 23
1 Overview
1.1 Backgroud
As a legendary creature with great and magical power, the dragon often appears in
various literature, art, lm and television works as well as architecture and monuments.
Almost all the world has legends about dragons, but dierent regions have dierent de-
scriptions of dragon images and their living habits.
In George Martin’s ”A Song of Ice and Fire”, the dragon has a strong body covered
with hard scales, sharp teeth, a long neck and a long tail. It ies with a pair of giant
wings like a batwing. The dragon is a kind of special mammal that lives in a cave and
spends most of its time alone. The dragon does not exist in the real world, so people’s
research on the dragon mostly focuses on the realms of history, literature and art. There
are not many special studies on the biological aspects such as the dragon’s physiological
characteristics, behavior, habits, diet, and interaction with the ecological environment.
1.2 Restatement of the Problem
If the ctional three dragons in The Song of Ice and Fire are still alive, we need to
analyze the characteristics, requirements, and interactions of the creatures
1. How does the weight of a dragon change according to age?
2. How does the calorie consumption and intake of the dragon change according to
age?
3. How large is the area required to support the dragon’s life under dierent climatic
conditions? What is the impact of climatic conditions?
4. How large a community is needed to provide dierent levels of assistance for the
dragons?
5. How does the dragons inuence the ecological environment? How severe is the
impact?
2 Assumptions and Justications
To simplify our problems, we make the following basic assumptions, each of which is
properly justied.
• Assumed that three dragons have the same characteristics in aspects of growth,
behavior, habit, diet and other aspects;
• Assumed that the growth of a dragon is restricted by its own genes and environ-
mental conditions, so it cannot grow indenitely, and its weight growth rate is a
linear weight loss function;
• Assumed that the weight of a dragon at birth is about 10kg, and it will weigh about
35kg one year later, 14000kg three years later.
Team # 1902727 Page 3 of 23
• Supposed that the dragon is an egg-laying mammal (similar to a platypus), whose
metabolism is similar to that of humans;
• Assumed that the basic metabolism of the average human takes up 70% of the
calories consumed in a day;
• Assumed that the body of the dragon is composed of water, protein, fat, carbo-
hydrate, mineral and etc, with the proportion of protein, fat and carbohydrate to
16
• Supposed that the dierence between the calorie intake and consumption of the
dragon is converted into the protein, fat and carbohydrate that make up the dragon’s
body;
• Assumed that the area of land needed by the dragon includes the area of predation
area, the area of drinking water area and the area of living area. The area of drinking
water area and living area is relatively small for the area of predation area, so the
area of land needed by the dragon is the area of predation area.
• Supposed in 4.3, the dragon lives in a simplied grassland ecosystem, in which the
biological community is only composed of microorganisms, dragons, sheep of the
same species and herbage. The dragon’s main food is sheep, and its only source of
calories is sheep.
• Assumed that the average weight of the sheep as dragon’s food source, and the
proportion of the portion of the edible part of the edible part of the diet, the
calories in the size of a 1kg of lamb and the calories of the lamb and the amount of
calories that are in the area of the grass, the average weight is about 100 kg, and
the average weight is about 50% of the food, and the amount of calories that is in
the 1kg is about 2030kcal;
• Assumed that the birth rate and natural death rate of sheep are constant, the daily
birth rate and natural death rate are 0.107% and 0.044% respectively.
Team # 1902727 Page 4 of 23
3 Notation
Abbreviation Description Unit
a Age of dragon year
w(a) Body mass of dragon at age of a kg
w
0
Body mass of dragon at birth kg
w
m
Maximum body mass of dragon kg
r(w) The weight increasing rate kg/year
q(a) Dragon’s daily expenditure at age of a kcal/day
q
0
Energy expenditure of a average male adult kcal/day
BMR
0
Basal metabolic rate of a average male adult kcal/day
u(a) Dragon’s daily intake at age of a kcal/day
s(a) Number of sheep a dragon need to eat per day day
−1
b Birth rate of sheep ock day
−1
d Natural mortality of sheep ock day
−1
s
m
The number of sheep the three dragons day
−1
G Scale of sheep ock under an ecological balance
g(t) The number of sheep after breeding for t days
g
0
Sensitive coecient of forage yield
h
i
(t) In t th month, grass yield kg/hm
2
Y
t
Ihe value of the i th meteorological element kg/hm
2
x
i
(t) meteorological element
y Grass Yield kg/hm
2
c
i
The animals calories of a type i ecosystem provides kcal × day/m
2
S
i
Type i ecosystem area required to sustain dragons m
2
f
i
The average biomass of type i ecosystems kg/m
2
F
i
The community size required kg
n
i
The net primary production(NPP)of type i ecosystem g × year/m
2
∆NP P
ij
The reduced NPP after type i degenerates into j g/year
∆F
ij
The reduced biomass after type i degenerates into j kg
4 Analysis and Modeling
4.1 Dragon’s weight growth equation based on Logistics Model
According to biological analysis, due to the limitation of food resources and environ-
mental conditions, the weight of an organism cannot grow indenitely, and its growth
rate (ie, the rate of weight gain) increases rst and then decreases with the increase of its
age[1], showing a growth retardation feature. Thus, we established a dragon weight gain
model based on the retardation growth model.
Assume that the growth rate of dragon’s weight r(w) is a function of the weight of
the dragon w(a) , a is its age. According to the previous analysis, r(w) should be a
decreasing function of w. For simplication, we assume that r(w) is a linear function of
w, r(w) = r
0
−sw . Under the constraints of food resources and environmental conditions,
the dragon’s maximum weight is w
m
, when w = w
m
, the dragon’s weight increase rate is
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