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March 7, 2013

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�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 9 / 21

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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)}.

�<� ([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

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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}.

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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

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êÆï�«~�n: XÛý�<��O��êO��.(Malthus)

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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�{¢O��.(Verhulst)

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�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 14 / 21

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�<� ([email protected]) êÆ�.: ïáêÆ�. March 7, 2013 14 / 21

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