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�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 9 / 21
êÆï�«~��: û<�N�S�Làn¶û<�����l�ELà, ���E�UNBü<, d��gCy1"�l���, 3à�?�W, ���l�<ê'û<õ, Òà<�À"�´XÛ�EÞà���ݺ3û<��¥"û<�N�âUS�Þà?P1kgÞà dW�û<êǑxk, �lêǑyk.G�sk = {(xk, yk)}.#NG�8Ü:
S = {(x, y) | x = 0, y = 0, 1, 2, 3;x = 3, y = 0, 1, 2, 3;x = y = 1, 2}.
�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 9 / 21
êÆï�«~��: û<�N�S�Làn¶û<�����l�ELà, ���E�UNBü<, d��gCy1"�l���, 3à�?�W, ���l�<ê'û<õ, Òà<�À"�´XÛ�EÞà���ݺ3û<��¥"û<�N�âUS�Þà?P1kgÞà dW�û<êǑxk, �lêǑyk.G�sk = {(xk, yk)}.#NG�8Ü:
S = {(x, y) | x = 0, y = 0, 1, 2, 3;x = 3, y = 0, 1, 2, 3;x = y = 1, 2}.ûüdk = (uk, vk), uk, vk©OǑEþû<êÚ�lê"�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 9 / 21
êÆï�«~��: û<�N�S�Làn¶û<�����l�ELà, ���E�UNBü<, d��gCy1"�l���, 3à�?�W, ���l�<ê'û<õ, Òà<�À"�´XÛ�EÞà���ݺ3û<��¥"û<�N�âUS�Þà?P1kgÞà dW�û<êǑxk, �lêǑyk.G�sk = {(xk, yk)}.#NG�8Ü:
S = {(x, y) | x = 0, y = 0, 1, 2, 3;x = 3, y = 0, 1, 2, 3;x = y = 1, 2}.ûüdk = (uk, vk), uk, vk©OǑEþû<êÚ�lê"#Nûü8Ü:
D = {(u, v) | 1 ≤ u+ v ≤ 2, u, v = 0, 1, 2}.
�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 9 / 21
êÆï�«~��: û<�N�S�Làn¶û<�����l�ELà, ���E�UNBü<, d��gCy1"�l���, 3à�?�W, ���l�<ê'û<õ, Òà<�À"�´XÛ�EÞà���ݺ3û<��¥"û<�N�âUS�Þà?P1kgÞà dW�û<êǑxk, �lêǑyk.G�sk = {(xk, yk)}.#NG�8Ü:
S = {(x, y) | x = 0, y = 0, 1, 2, 3;x = 3, y = 0, 1, 2, 3;x = y = 1, 2}.ûüdk = (uk, vk), uk, vk©OǑEþû<êÚ�lê"#Nûü8Ü:
D = {(u, v) | 1 ≤ u+ v ≤ 2, u, v = 0, 1, 2}.G�=£�§:
sk+1 = sk + (−1)kdk, sn+1 = (0, 0).�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 9 / 21
êÆï�«~��: û<�N�S�Làx
y
O 1 2 3
1
2
3
éuû<êǑm, �lêǑn, �E1p<, ¯K��)59)XÛ?
�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 10 / 21
êÆï�«~�n: XÛý�<��O��µ: .<�(UNFPA) 1625 1830 1930 1960 1974 1987 2000 2011<�(·) 5 10 20 30 40 50 60 70
�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 11 / 21
êÆï�«~�n: XÛý�<��O��µ: .<�(UNFPA) 1625 1830 1930 1960 1974 1987 2000 2011<�(·) 5 10 20 30 40 50 60 70¥I<� 1908 1933 1953 1964 1982 1990 2000 2008<�(·) 3.0 4.7 6.0 7.2 10.3 11.3 12.95 13.3
�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 11 / 21
êÆï�«~�n: XÛý�<��O��µ: .<�(UNFPA) 1625 1830 1930 1960 1974 1987 2000 2011<�(·) 5 10 20 30 40 50 60 70¥I<� 1908 1933 1953 1964 1982 1990 2000 2008<�(·) 3.0 4.7 6.0 7.2 10.3 11.3 12.95 13.3ïÄ<�Cz5Æ ��<�L¯O��<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 11 / 21
êÆï�«~�n: XÛý�<��O��êO��.(Malthus) 8 <�x0, O�Çr, Kk �<�xk = x0(1 + r)k.
�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 12 / 21
êÆï�«~�n: XÛý�<��O��êO��.(Malthus) 8 <�x0, O�Çr, Kk �<�xk = x0(1 + r)k.e�t�Ǒ<�Ǒx(t), O�ÇǑ~ê,
x(t+∆t)− x(t) = rx(t)∆t.
�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 12 / 21
êÆï�«~�n: XÛý�<��O��êO��.(Malthus) 8 <�x0, O�Çr, Kk �<�xk = x0(1 + r)k.e�t�Ǒ<�Ǒx(t), O�ÇǑ~ê,
x(t+∆t)− x(t) = rx(t)∆t.-∆t → 0,dx
dt= rx, x(0) = x0.
�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 12 / 21
êÆï�«~�n: XÛý�<��O��êO��.(Malthus) 8 <�x0, O�Çr, Kk �<�xk = x0(1 + r)k.e�t�Ǒ<�Ǒx(t), O�ÇǑ~ê,
x(t+∆t)− x(t) = rx(t)∆t.-∆t → 0,dx
dt= rx, x(0) = x0.Ïd, x(t) = x0e
rt.�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 12 / 21
êÆï�«~�n: XÛý�<��O��êO��.(Malthus) 8 <�x0, O�Çr, Kk �<�xk = x0(1 + r)k.e�t�Ǒ<�Ǒx(t), O�ÇǑ~ê,
x(t+∆t)− x(t) = rx(t)∆t.-∆t → 0,dx
dt= rx, x(0) = x0.Ïd, x(t) = x0e
rt.¯K: XÛ�Or? (§S: pop1.m)�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 12 / 21
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�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 13 / 21
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�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 13 / 21
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�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 13 / 21
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�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 13 / 21
êÆï�«~�n: XÛý�<��O��êO��.�A^9Û�5�19V± î³�/«<�ÚOêâ¬Ü·^u19V�[ \<��î³£¬���^uáÏ<�O�ýÿØÎÜ19V�õê/«<�O�5ÆØUýÿ��Ï�<�O�L§<�O�ÇrØ´~ê(Åìeü)
�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 13 / 21
êÆï�«~�n: XÛý�<��O��êO��.�A^9Û�5�19V± î³�/«<�ÚOêâ¬Ü·^u19V�[ \<��î³£¬���^uáÏ<�O�ýÿØÎÜ19V�õê/«<�O�5ÆØUýÿ��Ï�<�O�L§<�O�ÇrØ´~ê(Åìeü)
�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 13 / 21
êÆï�«~�n: XÛý�<��O�{¢O��.(Verhulst) �<�O�Ç(�Er�P~¼ê)
�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 14 / 21
êÆï�«~�n: XÛý�<��O�{¢O��.(Verhulst) �<�O�Ç(�Er�P~¼ê)
r(x) = r
(
1−x
xm
)
,
�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 14 / 21
êÆï�«~�n: XÛý�<��O�{¢O��.(Verhulst) �<�O�Ç(�Er�P~¼ê)
r(x) = r
(
1−x
xm
)
,Kdx
dt= rx
(
1−x
xm
)
, x(0) = x0.
�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 14 / 21
êÆï�«~�n: XÛý�<��O�{¢O��.(Verhulst) �<�O�Ç(�Er�P~¼ê)
r(x) = r
(
1−x
xm
)
,Kdx
dt= rx
(
1−x
xm
)
, x(0) = x0. =⇒ x(t) =xm
1 +
(
xm
x0− 1
)
e−rt
.
�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 14 / 21
êÆï�«~�n: XÛý�<��O�{¢O��.(Verhulst) �<�O�Ç(�Er�P~¼ê)
r(x) = r
(
1−x
xm
)
,Kdx
dt= rx
(
1−x
xm
)
, x(0) = x0. =⇒ x(t) =xm
1 +
(
xm
x0− 1
)
e−rt
.
0 50 100 150 200 250 300 350 400 4500
0.5
1
1.5
2
2.5
x
dx/d
t
Logistic Model
1700 1800 1900 2000 2100 2200 23000
50
100
150
200
250
300
350
400
450
year
Am
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Logistic Model
�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 14 / 21
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