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For office use only
T1
________________
T2
________________
T3
________________
T4
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Team Control Number
1924781
Problem Chosen
A
For office use only
F1
________________
F2
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F3
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F4
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2019
MCM/ICM
Summary Sheet
Here Be Dragons: The Ecology of Invasive Predators
Giant, fire-breathing dragons may exist in the A Song of Ice and Fire world created by
George R. R. Martin, but how would they fare in our world? To answer this question, we
sampled animal population data from representative regions around the world, fit a growth
function to dragons based on data from the problem and from the A Song of Ice and Fire
series, used modified allometric scaling laws, and Lotka-Volterra equations to build a model
of how the dragons would interact with environments here on Earth.
Our model works well because it allows for the dragons to grow forever, but also limits their
growth rate based on environmental factors in a meaningful way that is representative of real
ecological pressures. Not only will dragons need different amounts of land in different
climates, but they will also grow to different sizes based on the scarcity of prey. Also, our
model allows for individual dragons to die off due to starvation which lets the remaining
dragons have a better chance at survival. This flexibility in the model leads to some
interesting results, such as one or more dragons surviving in a certain area, but if the area is
tweaked slightly in either direction, the dragons die.
In a hot arid region, the dragons need 110,000 km
2
to survive and become stable at 86,000 kg
each. In an arctic region, the dragons need 42,000 km
2
and reach 3,500,000kg, and in a
temperate region, they need only 570km
2
and grow to 4,000,000 kg. As expected, the
dragons' ability to survive is directly correlated with the availability and scarcity of prey.
While this three-dragon scenario is not likely to happen on earth, the beauty of this model is
that it is dynamic enough to be highly generalizable. The hunting preference is robust enough
that it can be applied to any animal in any position on the food chain. Based on the habits of
any animal we can fit our model to imitate the hunting strategy it would take and apply it to
novel scenarios to learn more about its behavior.
By slightly modifying the model used, we showed that we were able to model the growth of
an invasive apex predator and see what will happen to the prey population given initial
conditions. This could be used as a precaution to stop harmful invasive species from entering
a sustainable and stable ecosystem.
Finally, using our analysis of dragons in different scenarios we recommended to George R.
R. Martin that he not send Daenerys and Drogon to fight the white walkers in the North, as
Drogon would not be able to meet his basal metabolic rate in such a sparse climate. We
informed George that in order for Drogon to survive he must always stay in warm temperate
regions where food is plentiful, and he can live sustainably within the ecosystem.
Team #1924781 Page 1 of 21
Contents
1 INTRODUCTION 2
2 ASSUMPTIONS AND JUSTIFICATIONS 3
3 DRAGON CHARACTERISTICS 4
3.1 Unrestricted Dragon Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2 Metabolism and Kleibler’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.3 Dragon Are Gluten Intolerant . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.4 ODE Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 DRAGONS IN DIFFERING CLIMATES 9
4.1 Warm Temperate: Sheep on Sheep on Sheep . . . . . . . . . . . . . . . . . . 9
4.2 Dry Arid: ...*Tumbleweed*... . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.3 Arctic: A Depressingly Small Amount of Polar Bears . . . . . . . . . . . . . 12
4.4 Generalizing the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5 SENSITIVITY ANALYSIS 15
6 WEAKNESSES AND FUTURE IMPROVEMENTS 17
6.1 Better Population Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6.2 Dragon-Dragon and Prey-Prey Interactions . . . . . . . . . . . . . . . . . . . 17
6.3 Scarcity Coefficient Ambiguity . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
7 INFEASIBILITIES OF DRAGONS’ EXISTENCE IN CONTEMPORARY WORLD 17
7.1 Thin legs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
7.2 Small wing area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
7.3 Immortality and unbounded growth . . . . . . . . . . . . . . . . . . . . . . . . 18
8 CONCLUSIONS 18
9 A LETTER TO GEORGE R.R. MARTIN 19
Team #1924781 Page 2 of 21
1 INTRODUCTION
We are required to analyze dragons’ characteristics, behavior, habits, diet and interaction with
their environment. Then build a mathematical model to analyze the ecological impacts the dragons
from Game of Thrones could introduce to our world. We decompose the problem into four sub-
problems:
• Build a model to estimate the growth of dragons.
• Build a model to estimate energy expenditures of dragons.
• Propose a hunting strategy for dragons.
• Base our model on Lotka–Volterra equations, modify them and apply to multi-prey situations.
In the first step, we seek to determine the scaling law of a dragon’s physique, using known data
to estimate its size at maximum growth rate.
In the second step, we seek to use a
3
4
allometric scaling power law to evaluate metabolic rate
of dragons at given body masses. We will use a modified version of the scaling law to improve the
accuracy additionally.
In the third step, we aim to propose a biased hunting strategy that is related to relative size of
the prey, population density, and climate.
Finally, based on the already matured predator-prey model, we consider multiple species in
different regions, justify the determination of several parameters, and put our model to the test.
Figure 1: The logic framework of our paper.
Team #1924781 Page 3 of 21
2 ASSUMPTIONS AND JUSTIFICATIONS
• Whenever there is a discrepancy between the books and the TV show we will
assume the TV show is correct. Because the problem references the television show
directly and not the book.
• Whenever there is a discrepancy between the world of Game of Thrones and
physics we will assume the world described by George RR Martin is correct. We
have to work within the rules given to us by the world creator. We chalk the discrepancy up
to magic.
• Given infinite resources the dragon will continue to grow indefinitely as mentioned
in the problem’s description.
• All animals in a species will die of natural causes at the same age unless they are
eaten by a dragon. This is a simplification
• In each species population there is a uniform spread of ages throughout the group.
If dogs live for ten years and there are one hundred dogs in a sample population, we assume
there are ten dogs who are between zero and one years old, ten dogs who are between one and
two years old, etc. In view of the fact that this simplification doesn’t affect the results much.
• Dragons do not have a preference for any type of animal meat over another. The
mass and population density of prey are the only parameters that matter in the dragons
hunting strategy. We made this assumption as the TV show doesn’t present the dragons’
food preference whatsoever.
• Drogon displays the maximum growth rate for a dragon in seasons 1 through 7
in Game of Thrones. The dragon has free access to roam as far as possible. We know
keeping dragons in captivity stunts their growth, this does not affect Drogon. We see from
the complaints of farmers that Drogon has access to livestock, which is essentially unlimited
food that he doesn’t have to hunt for. Therefore, we believe that it is reasonable to assume
that Drogon grew as fast as a dragon could reasonably expect to grow.
• Dragons, while they have the ability to grow much larger than other animals,
basal metabolic rates scale with mass as other animals metabolic rates do. We
are assuming that dragons are in the kingdom of Animalia and hence obey allometric scaling
laws.
• Although reptiles are mostly ectotherms, dragons breath fire, so we assume they
are warm blooded and that their internal body temperature is similar to other mammals.
• In our models the only interaction is between dragons and their prey. There are
no dragon-dragon or prey-prey interactions. We made this assumption because dragons don’t
appear to practice cannibalism and we believe because of the size dragons can grow to, prey-
prey interactions are insignificant compared to dragon-prey interactions.
• The birth and death rates that we assume in our models encapsulates all natural
factors including: perdition, disease, aging, and availability of resources.
Team #1924781 Page 4 of 21
• The current population of animals has already reached a steady state equilibrium.
This assumption follows from the fact that we are assuming there is no prey-prey interaction.
• The dragons are dropped into their environments today and are hatched as baby
dragons. This assumption enables us to present a more complete model.
• Each dragon has a preference to hunt something that is its own size rather than
something smaller or larger than itself. However, we assume a dragon can hunt some-
thing up to about 3 times as large as itself as a dragon can breathe fire. The reason that a
dragon would not prefer a larger prey is because there is a high energy cost to breathing fire.
• There is a change in the amount of food a dragon can eat in a day based on how
scarce food is. The amount this changes is dependent on the environment the dragon is
in. We made this assumption because the density of food has an inverse relationship with
hunting difficulty.
• Each species is uniformly distributed throughout the area specified in each sim-
ulation. This is a simplification, in a better model the distribution would be based on the
herd behavior of the species
• If the mass of a dragon drops to 60% of its previous maximum mass, the dragon
dies of starvation. Humans will die if they drop to 60% of their normal mass. Since dragons
do not stop growing, we assume at any point their max mass was at once their ”normal” mass.
• Dragons all grow at the same rate if they are in the same environment. This is a
simplification, there would likely be some random variation due to the different activity levels
of the dragons and different genetic makeup.
• Dragons are roughly 90% efficient when they eat. 10% of ingested mass becomes waste.
• The energy density of meat is 2500 kcal/kg for all animals This is a simplification, in
a perfect model we would find the energy density for all different types of meat.
3 DRAGON CHARACTERISTICS
3.1 Unrestricted Dragon Growth
Our first goal is to measure the growth rate of a dragon with infinite resources. It is stated in
the problem that dragons ”continue to grow throughout their life depending on the conditions and
amount of food available to them.” We know based on the show, however, that the rate of growth
is not unrestricted. After a year of growth we know that the dragons have gained between 20 to 30
kg even thought they certainly had access to ample food and were not held in captivity. In order
to find this growth rate we first gathered concrete data points on the size of Drogon as functions of
his age, the results of which are shown in table 1. [7]
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