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For ofﬁce use only

Team Control Number

1919022

Problem Chosen

For ofﬁce use only

2019

MCM/ICM

Summary Sheet

Dragons and Ecosystem: 1 + n

Meta-population Model

Abstract

Dragons are the virtual creatures in the ﬁctional television series Game of Thrones. It is

interesting and meaningful to consider what would happen if the three dragons were living

today. The purpose of this problem is to simulate the competition game when new species

are introduced into the ecosystem. And the result of the simulation can be used for reference

in the realistic situations.

Our paper propose the 1 + n Meta-population Model, where 1 refers to the Growth

Model Based on Energy Flow focus on the dragon’s growth and n refers to the Improved

n-population Meta-populaiton Model focus on the ecosystem.

We ﬁrst form the Growth Model Based on Energy Flow to represent the growth process

of dragons in the food chain. We calculate the energy expenditure using the Kleiber’s Law

and ﬁnd the relationship between the energy intake and dragon’s mass based on the energy

ﬂow in the food chain. Referencing the animals similar to dragons, we set the upper bound

of dragon’s mass as 1.17 × 10

kg and time-varying growth rates.

Then Improved n-population Meta-populaiton Model is generated to quantatively es-

timate the dragons’ inﬂuence on the populations in the ecosystem. Based on the Tilman

n-population Meta-population Model, we form the initial ecosystem including n kinds of

populations and then introduce dragons into it. We regard dragons as the interference in the

ecosystem. According to the amount and the abundance of living beings in the ecosystem,

we can analyze the inﬂuence of dragons. Besides, we further take human beings’ recon-

struction of environment into consideration, offering guidance for human beings.

After that, we apply our models to the different kinds of environments including temper-

ate deciduous forest, desert and ice sheet. By calculating, we know that an adult dragon per

should take in 2.4 ×10

J energy per month and the region must be larger than 1.8 ×10

to support three dragons.

We then analyze the ecological response when dragons are introduced in the ecosystem,

which conforms to strong population extinction mechanism and Odd-even Symmetry Law.

In addition, we conduct sensitivity analysis about the area of the habitat and the time-lag.

Finally, we discuss strengths and weaknesses of our models.

Keywords: Energy Flow; n-population Meta-population Model; Growth Model; Habitat-

occupancy Proportion; Ecosystem inﬂuence

Team # 1919022 Page 1 of 29

1 Introduction

In the ﬁctional television series Game of Thrones, based on the epic fantasy novel series A

Song of Ice and Fire, three dragons are raised by Daenerys Targaryen, the “Mother of Drag-

ons.” As warm-blooded animals, the dragons’ behaviors should follow the basic law of biol-

ogy. Besides, the dragons also have special characteristics such as the ability to breathe ﬁre,

longevity and so on, making them capable to greatly inﬂuence the ecosystem.

It is interesting and meaningful to consider what would happen if the three dragons

were living today. Based on the reasonable physical and biological assumptions of dragons’

characteristics, we do research on the requirements and ecological impacts of the dragons.

Not only can it offer guidance about how to maintain the ecological settings of the story, but

it also provides insights into the game in realistic ecology.

The game of ecology, as one of the most important problems in ecology, has been drawing

researchers’ attention for a long time. Verhulst et. al [1] proposed Logistic Growth Model

to simulate the population growth in an asset controlling system. Lotka and Volterra [2]

presented the predator–prey equations to study the predator–prey competition. Hanski et.

al [3] proposed Two-population Meta-population Model. However, none of the above works

take environmental factors and other populations into account at the same time. More-

over, no prior works have ever introduced dragons, or any kind of virtual creature, into the

ecosystem.

In our work, we propose the 1 + n Meta-population Model combined by two models to

simulate the growth of dragons and their inﬂuences on the ecosystem.

The remainder of this article is organized as follows. In Section 2, we put forward the

assumptions and symbols used in this paper. Section 3 shows the Growth Model Based on

Energy Flow. Section 4 presents Improved n-population Meta-population Model. In Section

5, we apply the models to the different kinds of environment. In Section 6, we present the

results of the simulation and make analysis on them. In Section 7, we discuss the strengths

and weaknesses of our models. And ﬁnally, we conclude the paper in Section 8.

2 Assumptions and Symbols

2.1 Assumptions

First and foremost, we make some basic assumptions and explain their rationales.

Assumption. 1 The dragon’s biological structure and behaviours are similar to the warm-blooded

animals.

This assumption is the prerequisite of our model. With this assumption, we can analyze

dragons’ characteristics based on the existing knowledge.

Assumption. 2 During the time scale we discuss in this paper, the earth’s ecological environment

keeps stable.

In our model, we don’t consider the incidents that may cause great climate changes such

as meteorite impacts or frequent volcanoes, since their probabilities are small.

Assumption. 3 The existing communities are climax communities and have reached the homeostasis.

Team # 1919022 Page 2 of 29

The existing communities on the earth, going through a long-time succession, have adapted

to the local climate and reached a stable state.

Assumption. 4 During the time scale we discuss in this paper, a dragon’s body function won’t

change as the dragon grows old.

Dragons live a long life.

Assumption. 5 Dragons are at the top of the food chain.

Considering the power of dragons, the dragons can have inﬂuence on all living beings

in the ecosystem while few living beings can compete with the dragon.

Assumption. 6 A dragon’s body function won’t change when it is hurt.

Dragons can resist tremendous trauma.

Assumption. 7 There is a upper bound for dragon’s body size.

With the limitation of gravity and biological structure, the body size of a dragon cannot

inﬁnitely increase.

2.2 Symbols

The symbols we deﬁne in this paper are shown in the Table. 1.

Table 1: Symbols mentioned in this paper

Symbol Deﬁnition Unit

expend

Energy expenditure J

Energy intake J

fire

Energy used in breathing ﬁre J

assimilate

Energy the animal assimilates J

AR Assimilation rate −

NAR Net assimilation rate −

M Mass of a dragon kg

q Metabolic rate J · kg

−1

· h

−1

σ Bone strength N · m

−2

NP P Net primary productivity J · m

−2

s Minimal area to support a dragon’s living m

S A dragon’s living scope m

Habitat-occupancy proportion of the population i −

Migration rate of the population i −

Mean death rate of the population i month

−1

η Reduction ratio of the habitat-occupancy −

ξ Reduction ratio of the population types −

 Time-lag month

µ Human beings’ impact factor −

3 Growth Model Based on Energy Flow

In this section, we present the Growth Model Based on Energy Flow to simulate the

growing process of a dragon, covering both energy expenditure and energy intake tasks.

The framework of the model is shown in Figure. 1. At ﬁrst, we calculate the dragon’s energy

expenditure using Kleiber’s Law, which considers two motion states and the ﬁre-breathing

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