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04_results.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{graphicx}
\usepackage{subcaption}
\end_preamble
\use_default_options true
\maintain_unincluded_children false
\language english
\language_package default
\inputencoding auto
\fontencoding global
\font_roman "default" "Times New Roman"
\font_sans "default" "Times New Roman"
\font_typewriter "default" "Times New Roman"
\font_math "auto" "auto"
\font_default_family default
\use_non_tex_fonts true
\font_sc false
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\font_sf_scale 100 100
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\use_package amsmath 1
\use_package amssymb 1
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\use_package esint 1
\use_package mathdots 1
\use_package mathtools 1
\use_package mhchem 1
\use_package stackrel 1
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\use_package undertilde 1
\cite_engine basic
\cite_engine_type default
\biblio_style plain
\use_bibtopic false
\use_indices false
\paperorientation portrait
\suppress_date false
\justification true
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\index Index
\shortcut idx
\color #008000
\end_index
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\paragraph_indentation default
\is_math_indent 0
\math_numbering_side default
\quotes_style english
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\html_be_strict false
\end_header
\begin_body
\begin_layout Section
Results
\end_layout
\begin_layout Subsection
Results of the Model of Dragon's Growth
\end_layout
\begin_layout Subsubsection
Determination of
\begin_inset Formula $\lambda$
\end_inset
and
\begin_inset Formula $\mu_{m}$
\end_inset
\end_layout
\begin_layout Standard
According to Eq.
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:c.3.1"
plural "false"
caps "false"
noprefix "false"
\end_inset
, the dragon's weight
\begin_inset Formula $y$
\end_inset
is a function of time
\begin_inset Formula $t$
\end_inset
.
As
\begin_inset Formula $A=281.6$
\end_inset
tons is a constant, there are only two undetermined parameters:
\begin_inset Formula $\lambda$
\end_inset
and
\begin_inset Formula $\mu_{m}$
\end_inset
.
By using the constraints given by the problem description:
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\begin{cases}
y(0)=10\\
y(1)=b & 30\le b\le40
\end{cases}
\]
\end_inset
\end_layout
\begin_layout Standard
we solve the value of
\begin_inset Formula $\lambda$
\end_inset
and
\begin_inset Formula $\mu_{m}$
\end_inset
using SciPy's scipy.optimize.fsolve function.
Because the result obtained by SciPy is related to the given initial value,
we traverse all the initial values using step size 10 from 1 to 1000 for
\begin_inset Formula $\lambda$
\end_inset
and
\begin_inset Formula $\mu_{m}$
\end_inset
, and then substitute the obtained solution into the original equation to
measure its accuracy, and finally find the optimal solution.
According to the range of
\begin_inset Formula $b$
\end_inset
, we take three special values 30, 35, and 40 to solve the parameters, and
the final result is shown in Tab.
\begin_inset CommandInset ref
LatexCommand ref
reference "L-res"
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
\align center
\begin_inset Caption Standard
\begin_layout Plain Layout
Solving results
\begin_inset Formula $\lambda$
\end_inset
and
\begin_inset Formula $\mu_{m}$
\end_inset
\end_layout
\end_inset
\begin_inset Tabular
<lyxtabular version="3" rows="4" 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" bottomline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $y(1)$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\lambda$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\mu_{m}$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
30
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
26.31
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
1895.42
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
35
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
22.42
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
2224.25
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
40
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
19.76
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
2523.83
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\begin_inset CommandInset label
LatexCommand label
name "L-res"
\end_inset
\end_layout
\begin_layout Plain Layout
\end_layout
\end_inset
\end_layout
\begin_layout Standard
From Tab.
\begin_inset CommandInset ref
LatexCommand ref
reference "L-res"
plural "false"
caps "false"
noprefix "false"
\end_inset
, we can conclude that when
\begin_inset Formula $A$
\end_inset
and
\begin_inset Formula $y(0)$
\end_inset
is a constant, the larger
\begin_inset Formula $y(1)$
\end_inset
, the smaller
\begin_inset Formula $\lambda$
\end_inset
, because the faster the growth rate, the shorter the lag time; the larger
\begin_inset Formula $y(1)$
\end_inset
, the larger
\begin_inset Formula $\mu_{m}$
\end_inset
, because the dragon can grow bigger in a short time, and then it will grow
faster when bigger, which forms a virtuous cycle.
\end_layout
\begin_layout Subsubsection
Determination of Growth Stages
\end_layout
\begin_layout Standard
Once
\begin_inset Formula $\lambda$
\end_inset
and
\begin_inset Formula $\mu_{m}$
\end_inset
are determined, we can draw the tangent of
\begin_inset Formula $y$
\end_inset
at the inflection point, which can be written as follows:
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
y=\mu_{m}(t-\lambda)
\]
\end_inset
\end_layout
\begin_layout Standard
Then we can derive the three essential points and stages of the growth curve
by
\end_layout
\begin_layout Itemize
the intersection of a vertical line which passes the intersection of the
tangent and the horizontal line
\begin_inset Formula $y=0$
\end_inset
and the original
\begin_inset Formula $y$
\end_inset
-curve,
\end_layout
\begin_layout Itemize
the intersection of the tangent and the original
\begin_inset Formula $y$
\end_inset
-curve,
\end_layout
\begin_layout Itemize
and the intersection of a vertical line which passes the intersection of
the tangent and the horizontal line
\begin_inset Formula $y=A$
\end_inset
and the original
\begin_inset Formula $y$
\end_inset
-curve.
\end_layout
\begin_layout Standard
When
\begin_inset Formula $b=40$
\end_inset
, the results of our solution are shown in Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "L-fig1"
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-curve-y_1=40.svg
scale 35
BoundingBox 0bp 0bp 980bp 720bp
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Groth curve of a dragon when
\begin_inset Formula $y(1)=40$
\end_inset
\end_layout
\end_inset
\begin_inset CommandInset label
LatexCommand label
name "L-fig1"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Form Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "L-fig1"
plural "false"
caps "false"
noprefix "false"
\end_inset
, we can conclude that, when
\begin_inset Formula $y(1)=40$
\end_inset
, the dragon's asymptotic weight value
\begin_inset Formula $A$
\end_inset
can be reached after nearly 280 years, which is a long time.
The periods of the three stages of a dragon are:
\end_layout
\begin_layout Enumerate
The nitial growth stage: 0 - 19 years old
\end_layout
\begin_layout Enumerate
The exponential growth stage: 19 - 131 years old
\end_layout
\begin_layout Enumerate
The steady growth stage:
\begin_inset Formula $>$
\end_inset
131 years old
\end_layout
\begin_layout Standard
The time when the dragon reaches maximum growth rate
\begin_inset Formula $\mu_{m}$
\end_inset
is around 52 years old, and it weighs 81,800.71 kg.
\end_layout
\begin_layout Subsection
Results of the Model of Dragon's Energy Expenditures
\begin_inset CommandInset label
LatexCommand label
name "subsec:Results-of-the"
\end_inset
\end_layout
\begin_layout Standard
By applying Dragon Hunting Algorithm, we keep the preset parameters unchanged,
run and output the results for further analysis:
\end_layout
\begin_layout Itemize
We simulate the age of dragon form
\begin_inset Formula $0$
\end_inset
to
\begin_inset Formula $500$
\end_inset
years old.
\end_layout
\begin_layout Itemize
The dragon hunts once every
\begin_inset Formula $t_{d}=$
\end_inset
2 days.
\end_layout
\begin_layout Itemize
We assume
\begin_inset Formula $y(1)=40$
\end_inset
and make use of the result of Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "L-fig1"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
If using other value of
\begin_inset Formula $y(1)$
\end_inset
, the results will be similar.
\end_layout
\begin_layout Itemize
The side of the square area is empirically set to
\begin_inset Formula $10$
\end_inset
meters to meet the dragon's hunting demand.
\end_layout
\begin_layout Itemize
The dragon starts hunting at position
\begin_inset Formula $(0,0)$
\end_inset
, which is the left-bottom of the square.
\end_layout
\begin_layout Itemize
Values of constant in Subsection
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Notations"
plural "false"
caps "false"
noprefix "false"
\end_inset
we use are shown in Tab.
\begin_inset CommandInset ref
LatexCommand ref
reference "L-values"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\end_layout
\begin_layout Subsubsection
Simulation Results
\end_layout
\begin_layout Standard
By running Algorithm
\begin_inset CommandInset ref
LatexCommand ref
reference "Z-algorithm"
plural "false"
caps "false"
noprefix "false"
\end_inset
, we get the output as Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "Z-En-year"
plural "false"
caps "false"
noprefix "false"
\end_inset
, Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "Z-2"
plural "false"
caps "false"
noprefix "false"
\end_inset
and Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Energy-percentage-of"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\end_layout
\begin_layout Standard
Energy expenditures of a dragon and preys eaten in two days vary with age
is shown in Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "Z-En-year"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
From Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "Z-En-year"
plural "false"
caps "false"
noprefix "false"
\end_inset
, we can see that:
\end_layout
\begin_layout Itemize
The energy a dragon need is also S-shaped.
This is because
\begin_inset Formula $E_{m}$
\end_inset
and
\begin_inset Formula $E_{t}$
\end_inset
increase as
\begin_inset Formula $y$
\end_inset
increases.
\end_layout
\begin_layout Itemize
The curve is steeper than the growth curve Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "L-fig1"
plural "false"
caps "false"
noprefix "false"
\end_inset
at the inflection age (52 years old).
This is because
\begin_inset Formula $E_{g}$
\end_inset
increases as
\begin_inset Formula $\mu$
\end_inset
increases, not
\begin_inset Formula $y$
\end_inset
, the former increases faster than the latter.
\end_layout
\begin_layout Itemize
A mature dragon needs 178 sheep and 470 cattle, which is 140, 000 million
calories of energy, per two days.
\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/Prey intake-age=501.svg
scale 30
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Energy expenditures of a dragon and preys eaten in two days
\begin_inset Formula $-age$
\end_inset
curve
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Z-En-year"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Minimum area required to support the three dragons vary with age is shown
in Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "Z-2"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
As for minimum area required to support the three dragons, according to
Eq.
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:3.c.24"
plural "false"
caps "false"
noprefix "false"
\end_inset
, we can determine the minimum area to support a dragon.
Then the minimum area of three dragons is three times of
\begin_inset Formula $S(age)$
\end_inset
obtained by Eq.
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:3.c.24"
plural "false"
caps "false"
noprefix "false"
\end_inset
, which is shown in Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "Z-2"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
We can see that:
\end_layout
\begin_layout Itemize
The trajectory is the same as that of Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "Z-En-year"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
This is because the area required is proportional to the preys eaten.
\end_layout
\begin_layout Itemize
We need at least about 14, 000 km
\begin_inset Formula $^{2}$
\end_inset
area to support three mature dragons.
\end_layout
\begin_layout Standard
\begin_inset Float figure
placement H
wide false
sideways false
status collapsed
\begin_layout Plain Layout
\align center
\begin_inset Graphics
filename figures/area-501.svg
scale 30
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Minmum area required to support the three dragons
\begin_inset Formula $-age$
\end_inset
curve
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Z-2"