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For office use only

Team Control Number

For office use only

190926 3

T1 ________________

F1 ________________

T2 ________________

F2 ________________

T3 ________________

Problem Chosen

F3 ________________

T4 ________________

F4 ________________

2019 Mathematical Contest in Modeling (MCM) Summary Sheet

Abstract

Dragon has long been a creature of mystery and power in our imagination, with many miraculous

abilities such as fire-breathing, ultra-strong defensiveness and extraordinary fly-ability. In the fantasy

novel A Song of Ice and Fire, three dragons helped the heroine Daenerys Targaryen to seize power

and achieve her ambition. Now our team is required to analyze their features, behavior, predation

and interaction with environment. A few questions are also need to be answered, including the food

intake and energy consumption of dragon, the area required to support the long-term existence of a

dragon, and its influence on the environment.

Firstly, we analyzed the general features of dragon, and made some basic assumptions. In the

novel, though the dragon is formidable, many stories about killing the dragons still spread in the

world. Therefore, we believe that dragons are not immortal creatures but still under the limitations of

physics, based on which we discussed the size and the classification of dragon, and we proposed a

reasonable assumption about the mechanism of its fire-breathing.

Secondly, we discussed in detail two different models to predict the energy consumption of

animals, the first one is an empirical model based on the mass of organism, whose performance is

limited for the ambiguity of dragon. Then we introduced in another model to estimate the metabolic

rate of dragon according to the maximum growth rate and the results are reliable and have been used

in the later research.

Thirdly, we build a discrete model to calculate the biomass of each component in an ecosystem.

Initially we set the tigers as the dominant creature in the model, by which we could make an analogy

between our model and real world to modify the parameter of our model. Then we replaced the tiger

with dragon to calculate the biomass of dragon. By virtue of the isometric relationship between the

dragon and tiger, we established a bridge between the abstract model and real world. Finally, we

successfully obtained the environmental capacity of dragon.

Then we changed the parameters systematically to simulate different conditions, by calculating the

biomass of dragon in our model under different temperature and solar radiation, we fitted the curve

of the biomass against environmental variables, from which many instructive results can be found.

Later on, we changed the structure of community and verified some basic mechanisms in the nature.

Simultaneously we made the analyses of error and sensitivity, the results show that our model is

robust and reliable.

Finally, we proposed many meaningful suggestions according to the results and drafted a letter to

the author of The Song of Ice and Fire, in which we gave some suggestions on the settings of the

novel to help depict the fantasy world more vividly.

Key words: Metabolic rate, Food web

Team #1909263  Page 1 of 20 
 
Contents 
I. Introduction ...................................................................................................................................... 2 
 
1 Background ............................................................................................................................... 2 
 
2 Our works.................................................................................................................................. 2 
 
II.  Assumptions about Our Object of Study ..................................................................................... 3 
 
1 The physical properties of dragons ........................................................................................... 3 
 
2 Estimation on the energy consumption of dragons when flying ............................................... 3 
 
3 Description of reasonable mechanism of fire breathing ........................................................... 3 
 
4  Estimation on the size of dragon .............................................................................................. 3 
 III. Models ............................................................................................................................................ 3 
 
1  Modeling on the energy consumption of dragons.................................................................... 3 
 
1.1 Energy Consumption Model based on Mass .................................................................. 4 
 
1.2 Energy Consumption Model based on Growth Rate ...................................................... 4 
1.2 Energy Consumption Model based on Growth Rate ............................................................. 5 
1.2 Energy Consumption Model based on Growth Rate ............................................................. 6 
1.2 Energy Consumption Model based on Growth Rate ............................................................. 7 
 
 
2 Modeling on the area of bio-system to sustain the dragon ....................................................... 8 
 
2.1 The Theoretical Foundation of our Model ...................................................................... 8 
 
2.2 Discrete Energy Flux Model ........................................................................................... 8 
2.2 Discrete Energy Flux Model ............................................................................................. ....9 
2.2 Discrete Energy Flux Model ................................................................................................ 10 
2.2 Discrete Energy Flux Model ................................................................................................ 11 
2.2 Discrete Energy Flux Model ................................................................................................ 12 
2.2 Discrete Energy Flux Model ................................................................................................ 13 
 
IV. Error Analysis and Sensitivity Analysis .................................................................................... 13 
 
1 Error analysis on the model of energy consumption .............................................................. 13 
1 Error analysis on the model of energy consumption...................................................................... 14 
2 Sensitivity Analysis ................................................................................................................ 14 
2.1 The Influence of Different Initial Biomass in the Model of Energy Flow ................... 14 
 
2.2 The Influence of Some Parameters in the Model of Energy Flow ............................... 14 
 
2.3 The influence of different time scale in the Model of Energy Flow ............................. 14 
2.3 The influence of different time scale in the Model of Energy Flow.................................... 15 
 
V. Evaluation of Models .................................................................................................................... 15 
 
1 Strengths ................................................................................................................................. 15 
 
 
2 Weaknesses ............................................................................................................................. 15 
 
 VI. Conclusions .................................................................................................................................. 15 
 
 
Ⅶ. References ................................................................................................................................... 15 
Ⅷ. A letter to George R.R Martin ................................................................................................ 16 
 
IX. Appendix ...................................................................................................................................... 18 

Team # 1909263 Page 2 of 20

I. Introduction

1.1 Background

1.1.1 The dragon in the real world

In the fictional television series Games of Thrones based on the epic fantasy novel A Song of

the Ice and Fire, the “Mother of Dragons”, Daenerys Targaryen raised 3 dragons named Drogon，

Rhaegal, Viserion. In the relevant descriptions of the novel, the dragon is born small about 10

kilograms like a cat which grows to roughly 30-40 kilograms after 1 year. They develop quickly

and continuous in their whole life depending on the conditions and the amount of food available to

them, are capable of fire-breathing and flying distantly at will, resistant to tremendous traumas,

such as severe cold or extreme heat, as their scales covering the whole body are harder than horns

which can protect them. Besides, the dragon has no fixed sex, neither male nor female, which won’t

be a part of our discussion. In short, the dragon is a large, fierce and serpent-like legendary creature

with two wings and two legs[1].

As a matter of fact, it’s just a mythical creature of European mythology, which doesn’t really

exist in the world, so we have to draw an analogy with some other animals living in the real

physical world to model successfully.

1.1.2 Description of the problems

As requested, our team should analyze the characteristics, habits, diet, and interactions with

the environment of the dragon in this paper.

According to our analyses, the key to solving the problem is to work out the energy consumption of

the dragon, specifically speaking, we must build an appropriate model to predict the energy

consumption of animals of different types and sizes. On this basis, we can further analyze the

energy intakes of dragons and their demands on environment.

The area needed to support the existence of the dragon is also required, which we interpret as

the environment capacity of the dragon in a certain ecosystem. Afterwards, some of the coefficients

in this model need to be modified to simulate minimum area under different circumstances, in

particular, we need to take temperature, illumination and other organisms into account to increase

accuracy.

Finally, based on the previous work, we can perform some quantitative or qualitative analysis

to predict possible phenomena in a world where dragons exist and give some advice on maintaining

the ecological foundation.

1.2 Our works

(1) Established two models respectively to predict the energy consumption of animals, discussed

their disadvantages and advantages and predicted the metabolic rate of dragons based on one of

them.

(2) Predicted the mass of adult dragons through the fitting of weight against height of different

birds.

(3) Built a hydrodynamic model to predict the energy consumption of dragons when breathing fire.

(4) Analyzed the material consumption of dragons when generating fuels used for fire-breathing.

(5) Calculated the amount of food intake and compared the results with relevant descriptions in the

novel.

(6) Established a discrete model on the basis of the energy flow in food chain, and estimated the

environment capacity of dragons in tropical grassland ecosystem by comparing with some known

statistical results.

(7) Attained the environment capacity of dragons under diverse circumstances by modifying

relevant parameters.

(8) According to the results of our models, wrote a letter to the author to give some suggestions on

maintaining the ecological foundation.

Team # 1909263 Page 3 of 20

II. Assumptions about Our Object of Study

2.1 The physical properties of dragons

We take for it as a kind of fierce reptiles with 2 legs, which is different from lizards or other

common reptiles. We draw an analogy between it and the king of cretaceous, the dinosaur since it

has more agile mobility and faster growth speed compared with gecko.

We assume that it’s small and weights 2.5-3kg like a cat when hatched, and after a year it grows

to 30-40kg, and after another one year, it grows into a giant with its body length over 8 meters and

its wingspan more than 20 meters. It may have a physiological structure like lizards and a hard but

smooth scale covering the whole body to protect against physical attack and sharp changes of

environment temperature. As what the grand novel says, it mainly lives on cooked meat with the

amount several times more than their weight to grow up promptly [1].

All in all, it’s an extraordinary creature of imagination, fierce and enormous, that can possibly

exist in the real physical world.

2.2 Estimation on the energy consumption of dragons when flying

Based on some empirical evidences, the birds in flight generally consume 3-15times more energy

than other animals in running. For example, the metabolic consumption of hummingbird when

flying rapidly is more than 10 times as they do at rest. Due to the lack of sample, it’s difficult to get

the specific quantitative relationship, so in our models, we roughly believe that the energy

consumption of dragons when flying is about 10 times of that at rest (Wikipedia).

2.3 Description of reasonable mechanism of fire breathing

Assuming that not only gas, but also the flammable mixed liquids produced by decomposing

protein and starch can be produced in the stomach of dragon. We might as well suppose the

substance used in fire-breathing is a kind of miscible liquids whose major ingredient is ethyl

alcohol which can be stored in large quantities in the compartments differentiated from the gastric

pouch as liquids are safer and the density of liquids is much bigger than that of gas. Moreover,

when breathing fire, liquids can be sprayed farther, burn longer and the calorific value is also much

higher than that of gas so that the fire can be used for hunting and burning the living animals [2-3].

2.4 Estimation on the size of dragon

According to the energy conservation relation in the growth process of animals, we have every

reason to believe that the growth of dragon is restrictive. Based on the relevant description in the

novel and the portrayal in the television plot, we believe that when the dragon grows up completely,

its body length is about 8 meters and its wingspan is nearly 20 meters. In the novel, it only takes the

dragon less than 3 years from hatch to be capable of fighting. Hereby, we make some assumptions

about its growth and apply them in the following study[1].

Ⅲ. Models

In this section, we proposed two models to analyze the problems progressively. First, we found a

model to predict the gross energy consumption of this mysterious creature according to the figure,

abilities and other characteristics which were roughly depicted in the novel. Then, basing on the

results of the previous model, we tried to release this huge animal into a given ecosystem. After

comprehensive research and calculation we obtained the minimum area of one particular kind of

ecosystem that required to sustain the long-term existence of one dragon. Finally, we made

necessary modification and supplementation to our model and simulating different conditions in

terms of temperature, local biological communities and other factors.

3. 1 Modeling on the energy consumption of dragons

To calculate the energy consumption of dragon quantitatively, initially we divided the energy

consumption into two parts. The normal energy consumption, which means the proportion of

energy whose way of consuming is identical to that of other creatures exist in real world. It is

calculated based on Resting Metabolic Rate, a concept that is wildly used in the study of

animals ’metabolism, similar to the term Basing Metabolic Rate (BMR) when related to human

beings. And the abnormal energy consumption, which mainly refers to the energy used in fire-

Team # 1909263 Page 4 of 20

breathing. According to the assumption in section 3, the dragon should have a tank filled with

Ethanol inside its body and a powerful spout to interspersion the combustive liquid. We may

calculate the kinetic energy of the erupted liquid, and estimate the mass of row materials consumed

3.1.1 Energy Consumption Model based on Mass

(1) Model Establishment

According to our common sense, the energy consumption largely depends on the size, or to be

more precise, the mass of an animal. It is also proved by a large set of statistical date. This relation

can be denoted by the following equation, where B representing the resting metabolic rate and m

representing the mass:

B= （1）

The lower case letter is a constant to modify the equation. α is a coefficient that always assigned

as . According to the results of empirical measurements, after regression analysis, most of the

sample points fall around the obtained Regression curve. However things are quite different when

the research objects extend to involve other animals of different structure. From figure 1[4], we can

find that animals belongs to different categories can be divided into several groups. Thus, the

constant c mentioned above will be totally different. In fact , it will lead to a ten times discrepancy

of the constant between different categories.

Figure 1

●Metabolic rate scales with mass of different Class. Huge gap still exists after Logarithm Operation,

obviously we cannot directly use it as prediction model.

(2) Analysis of the Result

The regression curve can be roughly categorized into two types, the endotherm and ectotherm.

And we shall not arbitrarily consider dragon as either of them. As has been clearly depicted in the

background material, its ontogenetic process is tremendously rapid and it is also nimble enough to

fight and fly, moreover, it can adapt itself to various enormous climates.

To sum up, though the mass-based model performs quite well for known creatures and has valid

thermodynamics and structural foundation. Estimating the energy consumption of dragon basing on

this model will easily result in enormous inaccuracy due to the ambiguity of its categorization.

3.1.2 Energy Consumption Model based on Growth Rate

(1) Model Establishment

1) Energy Consumption Model based on Growth Rate

The discussion above shows that the key problem is the ambiguity of the categorization of this

animal, which naturally leading to following questions: could we develop a model that is nothing to

do with the category of certain animal? After a sequence of research, we find that relevant study

had already been conducted [5] on which our further research is based.

The relation between metabolic rates and mass had been illustrated in 3.1.1 which provides a

quantitative equation (1), at the same time, empirical evidence shows that the max growth rate has

similar quantitative relationship with mass, which is denoted as follows：

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