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03_submodules.lyx
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03_submodules.lyx
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#LyX 2.3 created this file. For more info see http://www.lyx.org/
\lyxformat 544
\begin_document
\begin_header
\save_transient_properties true
\origin unavailable
\textclass article
\begin_preamble
\usepackage{amsmath}
\DeclareMathOperator*{\argmax}{arg\,max}
\DeclareMathOperator*{\argmin}{arg\,min}
\end_preamble
\use_default_options true
\maintain_unincluded_children false
\language english
\language_package default
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\cite_engine basic
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\index Index
\shortcut idx
\color #008000
\end_index
\secnumdepth 3
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\end_header
\begin_body
\begin_layout Section
Model Establishment
\end_layout
\begin_layout Subsection
The Model of Dragon's Growth
\end_layout
\begin_layout Standard
The organism has a common feature on it's growing speed, which can be described
as Slow-Fast-Slow.
Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "Fig: growth curve"
plural "false"
caps "false"
noprefix "false"
\end_inset
clarifies the growth process of the organism.
\end_layout
\begin_layout Standard
\begin_inset Float figure
placement H
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Graphics
filename figures/Growth Curve.png
lyxscale 30
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Sigmoid Form Curve of Growth Process
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Fig: growth curve"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
As we can see in Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "Fig: growth curve"
plural "false"
caps "false"
noprefix "false"
\end_inset
, the total growth of the organism shows an 'S' trend that includes three
stages: the initial growth stage, the exponential growth stage, and the
steady growth stage.
These three stages are determined by two crossover point, which the tangent
line of the inflection point intersects with
\begin_inset Formula $y=0$
\end_inset
and
\begin_inset Formula $y=A$
\end_inset
.
\end_layout
\begin_layout Standard
Richards equation can simulate these three stages by four parameters with
biological meanings: the maximum specific growth rate
\begin_inset Formula $\mu_{m}$
\end_inset
, the lag time
\begin_inset Formula $\lambda$
\end_inset
, the asymptotic value
\begin_inset Formula $A$
\end_inset
and the shape parameter
\begin_inset Formula $v$
\end_inset
, which can be written as below
\begin_inset CommandInset citation
LatexCommand cite
key "C-zwietering-1990-modeling"
literal "false"
\end_inset
:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
y=A\left\{ 1+v\cdot\exp(1+v)\cdot\exp\left[\frac{\mu_{m}}{A}\cdot\left(1+v\right)^{\left(1+\frac{1}{v}\right)}\cdot(\lambda-t)\right]\right\} ^{\left(-\frac{1}{v}\right)}\label{eq:c.3.1}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
In order to simplify the problem,
\series bold
we suppose the dragon's shape satisfying geometrically similar with time
going by
\series default
\begin_inset CommandInset citation
LatexCommand cite
key "C-giordano-2013-first"
literal "false"
\end_inset
.
This can be shown as:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
S=kV^{\frac{2}{3}}\label{eq:3.c.2}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $S$
\end_inset
is the surface area of the dragon,
\begin_inset Formula $V$
\end_inset
is the volume of the dragon and
\begin_inset Formula $k$
\end_inset
is a constant.
If Richards equation satisfies Eq.
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:3.c.2"
plural "false"
caps "false"
noprefix "false"
\end_inset
, we could yield
\begin_inset Formula $v=-\frac{1}{3}$
\end_inset
, and Eq.
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:c.3.1"
plural "false"
caps "false"
noprefix "false"
\end_inset
only has three parameters that need to be determined, and it transforms
to Bertalanffy equation.
Considering the requirements in the problem that dragon's weight is
\begin_inset Formula $10$
\end_inset
kg when hatched and
\begin_inset Formula $30-40$
\end_inset
kg after a year's growth, so we need another requirement.
We tend to determine this by estimating
\begin_inset Formula $A$
\end_inset
, which indicates the maximum weight of the dragon.
\end_layout
\begin_layout Standard
Since the dragon does not exist in real life, we use the data of real creatures
to analogize the weight of a mature dragon.
Tyrannosaurus Rex is a good choice whose shape is very similar to the dragon.
After our investigation, we found that the largest T.
rex weighs about 14.85 tons.
The latest T.
rex was found in Montana, USA, in 1987.
It is the most complete specimen found, and it has a skull about five feet
long, which is equal to about 1.5 m
\begin_inset CommandInset citation
LatexCommand cite
key "Z-T.Rex"
literal "false"
\end_inset
.
For the dragon, we refer to the description in
\shape italic
Game of Thrones
\shape default
,
\shape italic
the biggest (skull of dragon) was the size of a carriage.
\shape default
We set the carriage with a length of 4 m.
Then, according to geometrical similarity, we can derive:
\end_layout
\begin_layout Standard
\align center
\begin_inset Formula
\[
\frac{14.85\times10^{3}}{A}=\frac{1.5^{3}}{4^{3}}
\]
\end_inset
\end_layout
\begin_layout Standard
From the equation above, we yield
\begin_inset Formula $A=281.6$
\end_inset
tons.
\end_layout
\begin_layout Subsection
The Model of Dragon's Energy Expenditure
\end_layout
\begin_layout Standard
After clarifying how the dragon grows, we need to determine the energy that
the dragon needs in daily life.
According to the researches in bioenergetics, we divide the individual
biological energy flow into several categories, which is shown in Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "Z-energy distribution"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\begin_inset Note Note
status open
\begin_layout Plain Layout
插入一个文献
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Float figure
placement H
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Graphics
filename figures/energy distribution.svg
lyxscale 50
scale 50
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Flow Chart of Energy Distribution
\begin_inset CommandInset label
LatexCommand label
name "fig:Flow-chart-of"
\end_inset
\end_layout
\end_inset
\begin_inset CommandInset label
LatexCommand label
name "Z-energy distribution"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
To estimate the total energy intake of a dragon, we tend to determine net
energy.
By our analysis, net energy can be divided into five parts, which are:
energy for basic metabolism, energy for growth, energy for flight, energy
for breathing fire and energy for recovery of trauma.
As we can see them located in Tab.
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Energy-cost-activities"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Float table
placement H
wide false
sideways false
status open
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Energy Cost Activities We Need to Analyze
\begin_inset CommandInset label
LatexCommand label
name "tab:Energy-cost-activities"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="6" columns="3">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
energy cost activity
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
category
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
influencing factor
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
energy for basic metabolism
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
basal metabolism
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
weight
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
energy for growth
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
growth energy
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
time
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
energy for flight
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
activity metabolism
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
location of food
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
energy for breathing fire
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
activity metabolism
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
amount of food
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
energy for recovery of trauma
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
activity metabolism
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
proportional to the energy for basic metabolism
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
From the consuming side, the total energy
\begin_inset Formula $E_{n}$
\end_inset
for the dragon can be written as Eq.
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:3.c.Energy"
plural "false"
caps "false"
noprefix "false"
\end_inset
below:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
E_{n}=E_{m}+E_{g}+E_{f}+E_{b}+E_{t}\label{eq:3.c.Energy}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
Next, we will determine each five part of
\begin_inset Formula $E_{n}$
\end_inset
.
\end_layout
\begin_layout Subsubsection
Energy for Basic Metabolism
\end_layout
\begin_layout Standard
Animals produce energy by breathing while they live on earth, which is called
cellular respiration.
Even though an animal remains static in a certain place, they need the
energy for basic metabolism
\begin_inset Formula $E_{m}$
\end_inset
.
We can calculate
\begin_inset Formula $E_{m}$
\end_inset
by the chemical equation below:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
C_{6}H_{12}O_{6}+6H_{2}O+6O_{2}\rightarrow6CO_{2}+12H_{2}O+energy\label{eq:3.2.1.c}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
If we could determine the oxygen consumption, then we can get the energy
released by the cellular respiration.
\end_layout
\begin_layout Standard
Oxygen consumption
\begin_inset Formula $V_{o_{2}}$
\end_inset
increases progressively less with the increase in species' weight.
\begin_inset Formula $V_{o_{2}}$
\end_inset
/kg is larger in small animals than in large animals, for small mammals
will have a larger body surface area/body mass ratio than heavier mammals.
Dragon is a huge organism, so they may have less
\begin_inset Formula $V_{o_{2}}$
\end_inset
/kg than the human.
\end_layout
\begin_layout Standard
Let
\begin_inset Formula $V_{E}$
\end_inset
to be
\begin_inset Formula $V_{o_{2}}$
\end_inset
\begin_inset Formula $/$
\end_inset
(kg
\begin_inset Formula $\cdot$
\end_inset
min).
By referring to the literature
\begin_inset CommandInset citation
LatexCommand cite
key "C-mortola-1985-breathing"
literal "false"
\end_inset
, we can get that
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\xout off
\uuline off
\uwave off
\noun off
\color none
\begin_inset Formula $V_{E}$
\end_inset
\family default
\series default
\shape default
\size default
\emph default
\bar default
\strikeout default
\xout default
\uuline default
\uwave default
\noun default
\color inherit
for an adult is
\begin_inset Formula $150$
\end_inset
ml
\begin_inset Formula $/$
\end_inset
(kg
\begin_inset Formula $\cdot$
\end_inset
min) in average, and we also can get that
\begin_inset Formula $V_{E}$
\end_inset
for the mouse that is
\begin_inset Formula $600$
\end_inset
ml
\begin_inset Formula $/$
\end_inset
(kg
\begin_inset Formula $\cdot$
\end_inset
min) in average, almost four times of an adult.
Because the scale comparison between a dragon and an adult is similar to
that between an adult and a mouse, we can estimate
\begin_inset Formula $V_{E}$
\end_inset
for dragon is
\begin_inset Formula $37.5$
\end_inset
ml
\begin_inset Formula $/$
\end_inset
(kg
\begin_inset Formula $\cdot$
\end_inset
min).
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
E_{m}=\frac{\rho_{o}m_{d}V_{E}t}{6\mu_{o}}E_{o}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\rho_{o}$
\end_inset
means the air density of oxygen,
\begin_inset Formula $m_{d}$
\end_inset
means the weight of the dragon,
\begin_inset Formula $\mu_{o}$
\end_inset
represents the relative molecular mass of oxygen, and
\begin_inset Formula $E_{o}$
\end_inset
means the energy provided by
\begin_inset Formula $1$
\end_inset
mol oxygen through Eq.
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:3.2.1.c"
plural "false"
caps "false"
noprefix "false"
\end_inset
to compose ATP.
\end_layout
\begin_layout Subsubsection
Energy for Growth
\end_layout
\begin_layout Standard
The dragon's body mostly consists of bone and muscle, so the energy for
dragon growing can be estimated by the bone growth and muscle growth.
As we assume that the dragon's shape satisfying geometrically similar varying
with time, the axial growth rate for these two components remains the same
but becomes different in radial direction.
This process can be shown by Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "Z-muscle&bone"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Float figure
placement H
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Graphics
filename figures/growth energy.svg
scale 50
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Sketch of Muscle and Bone
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Z-muscle&bone"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Let
\begin_inset Formula $r_{b}$
\end_inset
be the external radius of bone,
\begin_inset Formula $r_{m}$