Jan. 30, 2011 2011. (1) Malthusin Model (The Exponential Law) Malthus (1798) proposed a mathematical...
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Transcript of Jan. 30, 2011 2011. (1) Malthusin Model (The Exponential Law) Malthus (1798) proposed a mathematical...
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淺談數學模型與生態學
清大數學系
許世壁
Jan. 30, 2011 2011高中教師清華營
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(1) Malthusin Model (The Exponential Law) Malthus (1798) proposed a mathematical
model which assume the rate of growth is proportional to the size of the population. Let be the population size, then
where is called per capita growth rate or intrinsic growth.
I. 影響人類的生態數學模型
)(tx
0)0(
,
xx
rxdt
dx
r
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Then 馬爾薩斯在其書 ” An Essay on the Principle of population” 提出馬爾薩斯人口論。其主張為
人口之成長呈幾何級數,糧食之成長呈算術級數。
The rule of 70 is useful rule of thumb.1% growth rate results in a doubling every 70 years. At 2% doubling occurs every 35 years. (since )
rtextx 0)(
7.02ln
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(2) Logistic EquationPierre-Francois Verhult(1804-1849) in 1838 proposed that the rate of reproduction
to proportional to both existing population and the amount of available resources.
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Let be the population of a species at time ,
Due to intraspecific competition
)(tx t
capacitycarry
rategrowth intrinsic
)0(
,1
0
2
K
r
xx
K
xrxaxrx
dt
dx
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Besides ecology, logistic equation is widely applied in
Chemistry: autocatalytical reactionPhysics: Fermi distributionLinguistics: language changeEconomics:Medicine: modeling of growth of tumors
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K
2
K
t0'' x
0'' x
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As
Robert May ( Ph.D in plasma physics) 1970
given 0
),(1
0
1
x
xfxrxx kkkk
,40 r
]1,0[]1,0[: f
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Period-doubling cascade:
Logistic map shows a route to chaos by period-doubling
2 period ,...569946.3
216 period ,568759.3...5644.3
28 period ,5644.3...54409.3
24 period ,54409.3...449.3
2 period ,449.33
point fixed a toconverges ,31
44
33
22
1
0
r
rr
rr
rr
rr
xrr k
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is called the universal number discovered by
Feigenbaum. The number is independent of
the maps, for example
...669.4lim1
1
nn
nn
n rr
rr
,1 21 kk xrx
.sin1 kk xrx
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http://en.wikipedia.org/wiki/Logistic_map
http://demonstrations.wolfram.com/LogisticMapOnsetOfChaos/
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If you zoom in on the value r=3.82 and focus on one arm of the three, the situation nearby looks like a shrunk and slightly distorted version of the whole diagram
The bifurcation diagram is a fractal (碎形 ):
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is chaotic if(i) Period three period (ii) If has a periodic point of least period not a power of
2, then “Scramble” set S (uncountable) s.t. (a) in S
(b) period point of
Chaos in the sense of Li and YorkeReference: Li (李天岩 , 清華1968) and Yorke,
Period three implies chaos, AMS Monthly (1975)
]1,0[]1,0[:),(1 fxfx kk
kk,f
,0yx
0|)()(|inflim
,|)()(|suplim
yfxf
yfxf
nn
n
nn
n
,Sx p f
2|)()(|lim
pfxf nn
n
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Sharkovsky ordering
If and f has periodic point of period Then f has a periodic point of period .
Shorkovsky Theorem(1960):
12222
725232
725232
9753
21
222
nn
qpq
p
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is chaotic on if(i) has sensitive dependence on initial
conditions.(ii) is topological transitive(iii) Periodic points are dense in
is topological transitive if for there exists such that
Chaos in the sense of Devaney
VVf : V
V
VWU ,0k VUf k )(
ff
f
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Fashion Dress, designed and
made by Eri Matsui, Keiko Kimoto, and Kazuyuki
Aihara (Eri Matsui is a famous fashion designer in Japan)
This dress is designed based on the bifurcation diagram of the logistic map
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This dress is designed based on the following two-dimensional chaotic map:
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In the mid 1930’s, the Italian biologist Umberto D’Ancona was studying the population variation of various species of fish that interact with each other. The selachisns (sharks) is the predator and the food fish are prey. The data shows periodic fluctuation of the population of prey and predator.
The data of food fish for the port of Fiume, Italy, during the years 1914-1923:
Lotka-Volterra Predator-Prey model
1914 1915 1916 1917 1918 1919 1920 1921 1922 1923
11.9%
21.4%
22.1%
21.2%
36.4%
27.3%
16.0%
15.9%
14.8%
10.7%
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He was puzzled and turn the problem to his colleague, Vito Volterra, the famous Italian mathematician. Volterra constructed a
mathematical model to explain this phenomenon.
Let be the population of prey at time . We assume that in the absence of predation, grows exponentially. The predator consumes prey and the growth rate is proportional to the population of prey, is the death rate of predator
)(tx t
d
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Clnln
yieldsabove gIntegratin
0 variablesof separationBy
.,,
0)0(,0)0(
,
,
***
***
**
*
***
00
*
*
y
yyyy
c
b
x
xxxxV(x,y)
dyy
yy
c
bdx
x
xx
yybx
xxcy
dx
dy
b
ay
c
dx
yyxx
xxcydycxydt
dy
yybxbxyaxdt
dx
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Periodic orbits in phase plane
*x
*y
x
y
),( ** yx
levelEnergy V(x,y)
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Independently Chemist Lotka(1920) proposed a
mathematical model of autocatalysis
Where is maintained at a constant concentration . The first two reactions are autocatalytic. The Law of Mass Action gives
Independently Chemist Lotka(1920)
.
,
32
21
ykxykdt
dy
xykaxkdt
dx
a
BYYYXXXAkkk 321
,2,A
A
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Classical Lotka-Volterra Two-Species Competition Model
tscoefficienn competitio are,
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large is ,small is
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small is ,large is
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ncompetitioWeak
small are ,
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ncompetitio Strong
large are ,
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We assume: has same intrinsic growth rateIn the absence of , win over .In the absence of , win over .In the absence of , win over .
Competition of Three Species(Robert May 1976) 剪刀、石頭、布
321 ,, xxx r
1x
2x
3x
1x
1x
2x2x
3x
3x
1x
2x
3x
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0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-0.2
0
0.2
0.4
0.6
0.8
1
1.2
x1
x2
x3
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Thank you for your attention.