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©³¤U§Ú­Ì­n»¡©ú¡A¦b¶Ç¬V¯fªº¶Ç¼½¼Ò¦¡¤¤¤]·|¥X²{ Logistic ¼Ò«¬¡C°²³]¦³¤@ºØ¶Ç¬V¯f¡A±µÄ²«K·|·P¬V¡A¦Ó¥B·P¬V«á«K¯à§K¬Ì¥Ã¤£¦A¥Ç¡A¦]¦¹¦b®É¶¡ t¡A¬Y«°¥«ªº©Ò¦³¤H¤f M ¤¤¡A¥i¤À¦¨¤w·P¬V¤H¤f P(t) »P¥¼·P¬V¤H¤f M-P(t)¡A³]

\begin{displaymath}
\left\{
\begin{array}{rcll}
I(t) &=& \frac{P(t)}{M} & (\mbox...
... 0.0pt plus0.2pt minus0.1pt{\MhQ\char 48}})
\end{array}\right.
\end{displaymath}

±µµÛ°²³] £b ¬O¦b¦¹«°¥«¤¤¡A¨C¤H¨C¤Ñ¥­§¡±µÄ²ªº¤H¼Æ¡A«h¯f¤H¡]·P¬VªÌ¡^¨C¤Ñªº¼W¥[²v¬°

¨C­Ó¯f¤H¥­§¡¨C¤Ñ±µÄ²ªº°·±dªÌ¤ñ²v ¡Ñ Á`¯f¤H¼Æ ¡× $\big( \eta \times S(t) \big) \times \big( M \times I(t) \big)$

©Ò¥H¨Ï¥Î³sÄò¼Ò«¬¡A´N±o¨ì

\begin{eqnarray*}
&& (M \times I(t))' \; = \; \eta \cdot S(t)\cdot M\cdot I(t) ...
...Longleftrightarrow && I'(t) \; = \; \eta \cdot I(t)\cdot(1-I(t))
\end{eqnarray*}


³o¬O¤@­Ó Logistic ¼Ò«¬¡C

²ßÃD¡G
­Y¦Ò¼{ $\lambda= \frac{\eta}{M}$¡A»¡©ú¦¹¼Ò«¬¥i¥H§ï¼g¦¨§Ú­Ì§ó¼ô±xªº¼Ë¤l¡G

\begin{displaymath}
P'(t)=\lambda P(t)(M-P(t))
\end{displaymath}

(Hint: $P(t) = M\cdot I(t)$.)

¼g¦¨ I(t) ªº¦n³B¬O«°¥«¯u¥¿ªº¤H¤f¼Æ M ®ø¥¢¤F¡A¤ñ¨ÒªºÅܤƤ~¬O­«­nªº¡A©Ò¥H¥Ñ«e³¹ªº¤ÀªR¡A§Ú­Ìª¾¹D¸Ñ¦±½u I(t)¡]·íµM 0<I(t)<1¡^¬O¤@­Ó S «¬¦±½u¡A·í $I(t)=\frac{1}{2}$¡]§Y¥b¼Æ¤H¤f³Q·P¬V®É¡^¡A¬Ì±¡³Ì²r¯P¡C

²ßÃD¡G
(1) ³] $I(0)=\frac{1}{1000}$¡Aµø £b ¬°°Ñ¼Æ¡A¨D I(t)¡C
(2) ¨D¬Ì±¡³ÌÄY­«¡]§Y $I(t)=\frac{1}{2}$¡^ªº®É¶¡ t*¡C
(3) »¡©ú £b ¤j¤p»P t* ¬ÛÀ³ÅܤƪºÃö«Y¡C
¡]Ans. (1) $I(t)=\frac{1}{1+999e^{-\eta t}}$¡F (2) $t^*=\frac{\ln999}{\eta}$¡F(3) ¤Ï¤ñ¡^¡C

µù¡G³o»¡©ú¦b«°¥«©Î¤H¤f±K¶°³B¡A¥Ñ©ó £b ­È¸û¤j¡A¶Ç¬V¯f¬Ì±¡ªºÂX´²µ{«×»·§Ö©ó¶m§ø¡A¦]¦¹³£¥«ªº¤½½Ã³¡ªù¦b±±¨î¬Ì±¡®É¡A®É¶¡À£¤O§ó¤j¡C

²ßÃD¡G
¥x¥_¥«µo¥Í¶Ç¬V¯f¡A¨ä £b ¥Ñ¸ê®Æ¤¤¦ô­p¬° 1.7¡A¥x¥_¥«¤H¤f³]¬° 250 ¸U¤H¡A°²³]±w¯f¤H¼Æ¦b 1 ¤ë 15 ¤é®É¬° 5 ¤H¡A½Ð°Ý¤°»ò®É­Ô¬O±w¯f¤H¼Æ¼W¥[³Ì§Öªº®É¨è¡H¤S¤°»ò®É­Ô¦¹¶Ç¬V¯f·|¶i¤J¡u§ÀÁn¡v¡]°²³]©w¸q¬° $\frac{999}{1000}$ ¤H¤f³£±o¹L¦¹¯f¡H¡^
¡]Ans. 1 ¤ë23¤é¡F1 ¤ë27¤é.¡^

 
¹ï¥~·j´MÃöÁä¦r¡G
¡DLogistic¼Ò«¬
¡D¥­§¡­È©w²z
 
«D¥Ã¤[§K¬Ìªº¶Ç¬V¯f

·íµM¤£¬O©Ò¦³ªº¶Ç¬V¯f³£¯à¥Ã¤[§K¬Ì¡A¯f¤H¦bªv¡«á¡A¦³¥i¯à¦A­«·s·P¬V¯f±¡¡A³o®É§Ú­Ì»Ý­n­×§ï¼Ò«¬¬°

\begin{displaymath}
\big(M\cdot I(t)\big)'= \big(\eta\cdot S(t)\big)
\big(M\cdot I(t)\big) - \sigma \cdot M\cdot I(t)
\end{displaymath}

¨ä¤¤ £b ¦P«e¡A£m ªí¥Ü¨C¤éªv¡ªº¤H¼Æ¤ñ²v¡A¦]¦¹ $\frac{1}{\sigma}$ ªí¥Ü³oºØ¶Ç¬V¯fªº¡]¥­§¡¡^¯f´Á¡]¥J²Ó·Q·Q¬°¤°»ò¡H¡^¡C§Ú­Ì±o¨ì¤@­Ó·sªº¤èµ{¦¡

\begin{displaymath}
I'(t)=\eta I(t) \big(1-I(t)\big)-\sigma I(t)
\end{displaymath}

­ì¦¡¥i¥H§ï¼g¦¨

\begin{eqnarray*}
I'(t) &=& (\eta -\sigma)I(t)-\eta I(t)^2 \\
&=& \eta \cdot I(t)\cdot \Big( \big( 1- \frac{\sigma}{\eta} \big) -I(t) \Big)
\end{eqnarray*}


³o¤]¬O Logistic ¼Ò«¬¡C°ß¤@ªº¨Ò¥~¬O $1-\frac{\alpha}{\sigma}=0$ ¡]§Y $\sigma=\eta$¡^®É¡C

³o¸Ì¥X²{¤@­Ó­«­nªº«ü¼Ð¡G±µÄ²¼Æ £]¡A©w¸q¦¨

\begin{eqnarray*}
\beta \;\; \equiv \;\; {\frac{\eta}{\sigma}} &=&
(\mbox{ {\Mb...
...t{\MaQ\char 65}\hskip 0.0pt plus0.2pt minus0.1pt{\MbQ\char 98} }
\end{eqnarray*}


¦Ó¥B $1 - \frac{\sigma}{\eta} = 1- \frac{1}{\beta}$¡C

²ßÃD¡G
»¡©ú I(t) ªº¸Ñ¥i¼g¦¨

\begin{displaymath}
\left \{
\begin{array}{cl}
\frac{1}{\frac{\beta}{\beta-1}} ...
...{1}{\eta t+\frac{1}{I(0)}} & \quad \beta=1
\end{array}\right.
\end{displaymath}

§Ú­Ì¨Ì £] ­È¤À¦¨¤TºØ±¡ªp¨Ó°Q½×¡G

(1) $\beta <1$¡]¯f´Á¤¤¥­§¡±µÄ²¤H¼Æ¤Ö©ó 1 ¤H¡^¡C

«h $1 -\frac{\sigma}{\eta}=1-\frac{1}{\beta} <0$¡A¹Ï§Î¦p¤U¥ª¹Ï¡A ³oªí¥Ü§Y¨Ï­ì¨Ó¦³ 100 % ªº¤H±w¯f¡]I(0)=1¡^¡A¦ý¬O¥Ñ©ó $\beta <1$¡A ¯f¤H³Qªv¡«á¡A¦A­«·s¬V¯fªº¾÷·|¤£¤j¡AºCºCªº©Ò¦³¤H´N±d´_¤F¡C



$\beta <1$ ¡B $\beta = 1$ ¡B $\beta >1$

(2) $\beta = 1$¡]¯f´Á¤¤¥­§¡±µÄ²¤H¼Æ¬° 1 ¤H¡^¡C

³o¬OÃä¬É±¡ªp¡C¥Ñ¹Ïª¾¡A©Ò¦³¤H³£·|±d´_¡A¥u¬O±d´_ªº³t²v¬Ý°_¨Ó¤ñ(1)ªº±¡ªp­n½wºC¡C

(3) $\beta >1$¡]¯f´Á¤¤¥­§¡±µÄ²¤H¼Æ¶W¹L 1 ¤H¡^¡C

³oªí¥Ü¬Ì±¡¤´µM¦b¦³®Ä¦aÂX´²¡A¦Ó¥B³Ì«á±w¯f¤ñ²v·|ÁÍ©ó¬Y¤j©ó 0 ªºÃ­©w­È $1-\frac{1}{\beta}$¡A¤£·|®ø¥¢¡C

¦]¦¹¬°¤F®ø·À¬Ì±¡¡A­«­nªº¬O­n±N £] ­È½Õ¦¨ <1¡A¦ý $\beta =\frac{\eta}{\sigma}$¡A¦]¦¹³Ì¦n¯à

(1) ´î¤p £b¡A¥ç§Y¨Ï¥­§¡¨C¤é±µÄ²¼ÆÅܤ֡A©Ò¥H¯f¤H»Ý­n¹jÂ÷¡C
(2) ¥[¤j £m¡A¥ç§YÁYµu¶Ç¬V¯fªº¯f´Á¡AÄ´¦p»¡µo©ú·sÃÄ¡A·sªvÀøªkµ¥µ¥¡C

³o¨Ç³£»P±`Ãѧ¹¥þ²Å¦X¡C

¥t¥~¡A³o¤@ºØ¡«á¤£·|¦A·P¬Vªº¶Ç¬V¯f¡A¦pªG¦b¤HÃþªÀ·|¤¤¤w¸g¾ú¥v±y¤[¡A¨º»ò¥¦³q±`»P¤HÃþ¤§¶¡¤w¸g¹F¦¨¬YºØ¥­¿Å¡A³oºØ¯e¯fªº¯S¼x¬O¥¦¦bªø´Á¬y¦æªº°Ï°ì¤º¬J¤£·|­P©R¤]¤£·|µ´¸ñ¡]³o´N¬O $\beta >1$ ªº±¡ªp¡A·Q·Q´¶³q©Ô¨{¤l¡A·P«_µ¥¡^¡C¦ý¬O¡A¦pªG³oºØ¯e¯f³Q¶Ç¬V¨ì±q¥¼±µÄ²¥¦ªº¤H¸s¤¤¡A¥i¯à¾É­PÄY­«ªº¬Ì±¡¡A¬Æ¦Ü§ï¼g¾ú¥v µù1 ¡C

   
 
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«e­±´¿¸g´£¨ì¨âºØ¶Ç¬V¯f¼Ò«¬¡A²Ä¤@ºØ¼Ò«¬¡A±N©Ò¦³¤H¤À¬°¥¼³Q¶Ç¬VªÌ»P¤w¶Ç¬VªÌ¡A¤¤¶¡¨S¦³¤°»òªvÀøµ{§Ç¡A¨ì³Ì«á©Ò¦³¤H¤f³£·|¬V¯f¡]¦]¦¹¤j·§¤£¯àºÙ¤§¬°¯f¡^¡C²Ä¤GºØ¼Ò«¬¡A³B²zÃþ¦ü·P«_¡A©Ô¨{¤lªº¶Ç¬V¯f¡A¶Ç¬V¯fªv¡«á¨Ã¨S¦³§K¬Ì¤O¡A¦]¦¹¦b¤½¦@½Ã¥Í¤W¥²¶·¬I¦æ¬YºØµ{§Ç¡A±±¨î¶Ç¬V¯fªº¶Ç¼½¡C

¥»¸`­n³B²zªº¬O³Q¥t¤@ºØ±¡ªp¡G³\¦h¶Ç¬V¯f¡A¦p·ò¯l¡A¤Ñªá¦b±w¯fªv¡«á¡A¦³«Ü°ªªº§K¬Ì¤O¡A¦]¦¹³Qªv¡ªº¤H»P¥¼³Q¶Ç¬VªÌ¤£¯àµø¬°¦P¤@ºØ¸sÅé¡]¦p²Ä¤GºØ¼Ò«¬¡^¡C¦]¦¹§Ú­Ì¥²¶·±N©Ò¦³¤H¤À¦¨¤TºØ¡G°·±dªÌ¡]¤H¤f¤ñ¨Ò x(t)¡^¡A¯f¤H¡]¤H¤f¤ñ¨Ò y(t)¡^¡Aªv¡ªÌ¡]¤H¤f¤ñ¨Ò z(t)¡^¡C´N¹³²Ä¤GºØ¼Ò«¬¡A³o¨Ç¨ç¼ÆÀ³º¡¨¬¤U¦C·L¤À¤èµ{²Õ¡C

\begin{displaymath}
\left\{
\begin{array}{l}
x'=-\alpha xy\\
y'=\alpha xy-\beta y\\
z'=\beta y
\end{array}\right.
\end{displaymath}

¨ä¤¤¡A 0 < x(t), y(t), z(t) < 1¡A x(t)+y(t)+z(t)=1¡A$\alpha>0$ ¬O¨C¤H¨C¤Ñ±µÄ²ªº¤H¼Æ¡A$\beta > 0$ ¬O¨C¤Ñªv¡ªº¤ñ¨Ò¡A¦p«e¡A $\frac{1}{\beta}$ ¥i¬Ý¦¨¥­§¡¯f´Á¡A¨Ã©w¸q¯f´Á¥­§¡±µÄ²¼Æ $\eta= \frac{\alpha}{\beta}$¡C

²ßÃD¡G
°Q½× x(t), y(t), z(t) µ¥©ó 0 ©Î 1 µ¥Ãä¬É±¡ªpªº¸Ñ¤Î¨ä·N¸q¡C

¨Ò¡G¥x¥_¥«µo¥Í¬y¦æ©Ê·P«_¡]§K¬Ì´Á¸ûªøªº¯f¡^,¥Ñ¤½½Ã¸ê®Æª¾¹D¡A$\alpha =2 $¡A¥B $\beta =0.4 $¡A½Ð°Q½×¦¹¶Ç¬V¯f¤§ÂX´²¦æ¬°¡C

±µÄ²¼Æ $\eta =\frac{\alpha}{\beta}=5$¡C¦ý¬O¦b³o­Ó¼Ò«¬ùØ¡A§ó¦³·N¸qªº«ü¼ÐÀ³¬O¡]¶}©l®É¡^¯f´Á¥­§¡±µÄ²¤§°·±d¤H¼Æ¡]¼ÈºÙ¦³®Ä±µÄ²¼Æ¡^¡A 5 x x(0)=5x0¡Aª½Ä±¤W¥i²q´ú¦pªG 5 x0<1¡A«h¶Ç¬V¯fÀ³¸Ó¸ûÃøÂX´²¶}¨Ó¡C

³] z0=0¡A¥O x0=0.99, 0.2, 0.1¡A¦U¹ïÀ³©ó¦³®Ä±µÄ²¼Æ >1, =1, <1 ªº±¡ªp¡A¥Î¼Ú©Ôªk§@¹Ï±o¡]¬õ½u¬° x(t)¡Aºñ½u¬° y(t)¡AÂŽu¬° z(t)¡^¡G



¥Ñ¥ª¦Ü¥k x0=0.99¡Ax0=0.2¡Ax0=0.1

ª`·N¨ì·í $x_0\leq \frac{1}{5}=0.2$ ®É¡A¨ü¶Ç¬V¤H¼Æ y(t) ªGµMªu¸ô»¼´î¡A¦Ó x0=0.99 ®É¡Ay(t) ¥ý¤W¤É¦A¤U­°¡AÅã¥Ü¶Ç¬V¯fÂX´²ªºÁͶաC·íµM¤@¶}ÀY´N°²³]¡A¥¿·P¬VªÌ¦û¤F¤j³¡¤À¤ñ¨Ò¦ü¥G¬O¤@¥ó¯îÂÕªº¨Æ±¡¡A¥¿±`ªº±¡ªpÀ³¸Ó¬O x0 ±µªñ 1¡Ay0 ±µªñ©ó 0¡A¦]¦¹¬y¦æ©Ê·P«_ªºÂX´²¬OµLªkÁקKªº¡C

²ßÃD¡G
¥Î±`ÃѸÑÄÀ¬°¤°»ò·í y0 ¤j®É¡Ay(t) ·|»¼´î¡C

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©³¤U¬O¦b¬Û¦P°_©l±ø¥ó x0=0.99¡Ay0=0.01 ®É¡A¤£¦Pªº±µÄ²¼Æ $\eta=2$¡]¥ª¹Ï¡^¡A$\eta=16$ ¡]¥k¹Ï¡^©Ò¹ïÀ³ªº¬Ì±¡¦±½u¹Ï¡A§Ú­Ìµo²{±µÄ²¼Æ¶V¤j¡A¬Ì±¡¶Ç¬Vªº½d³ò¶V¤j¡A³t«×¶V§Ö¡C½Ð¥Î±`ÃÑ»¡©ú¤§¡C



¥Ñ«e¨Ò¹Ï§Î°Q½×¡A§Ú­Ìµo²{

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(2) ¡]°²³] $\eta >1$¡^
$1>x_0>\frac{1}{\eta}$¡A«h y(t) ·|»¼¼W¦A»¼´î¡]¶Ç¬V¯fÂX´²¡^¡C

$x_0\leq \frac{1}{\eta}$¡A«h y(t) »¼´î¡]¶Ç¬V¯f¤£ÂX´²¡^¡C

©³¤U§Ú­Ì§Q¥Î²³æªº©w©ÊÆ[¹îªk¡A¨Ó¸ÑÄÀ³o­Ó²{¶H¡C

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(i) ¡]¦b¶Ç¬V¯f´Á¶¡¡A¥¼³Q·P¬Vªº¤H·íµM·|»¼´î¡^
¥Ñ©ó $x'=-\alpha xy $¡A¦Ó¥B x(t), y(t)> 0¡C©Ò¥H x(t) ·|¤@¸ô»¼´î¨ì¬Y¥­¿Å¤ñ¨Ò $x_{\infty}$¡A·íµM³o¸Ì¤£¯à±Æ°£ $x_{\infty}$ ¥i¯àµ¥©ó 0 ªº¥i¯à©Ê¡C

(ii) ¡]¦]¬°¦³§K¬Ì¤O¡Aªv¡ªº¤H¼Æ¤@¸ôÃkª@¡^
$z'=\beta y$¡A¦Ó¥B $z(t)\leq 1$¡C©Ò¥H z(t) ¤@¸ô»¼¼W¨ì¬Y¥­¿Å¤ñ¨Ò $z_{\infty}$¡C

(iii) ¡]³Q¶Ç¬V¤H¼Æ¤]·|ÁÍ©ó¥­¿Å¡^
³o¬O¦]¬° x(t)+y(t)+z(t)=1¡A©Ò¥H $y_{\infty} =1-x_{\infty} - z_{\infty}$¡C

(iv) ¡]¦pªG¨ü¶Ç¬VªÌ©l²×¤£µ´¸ñ¡A¨ºªv¡ªÌ¤£¥i¯àÁÍ©ó¥­¿Å¡^
¥Ñ©ó $z'(t) =\beta y(t) \rightarrow \beta y_{\infty}$¡A©Ò¥H¦pªG $y_{\infty}\neq 0$¡A«h z'(t) Áͪñ©ó¤@¥¿ªº±`¼Æ¡A±q´X¦óª½Ä±ª¾¹D z(t) ¤£¥i¯àÁͪñ©ó¤@©w¼Æ $z_{\infty}$¡A³o»P (ii) ¥Ù¬Þ¡A©Ò¥H $y_{\infty}$ «D±oµ¥©ó 0 ¤£¥i¡C

¥Ñ(i)-(iv)¡A§Ú­Ìª¾¹D y(t) µL½×¦p¦ó·|Áͪñ©ó 0¡C

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§Q¥Î¥­§¡­È©w²z§ó¸Ô²Ó¦a»¡©ú¤W­±(ii)¤¤ªº´X¦óª½Ä±¡C§Y

¡u­Y z'(t) Áͪñ©ó¤@¥¿±`¼Æ¡A«h z(t) ¤£¥i¯àÁͪñ©ó¤@©w¼Æ $z_{\infty}$¡C¡v

(2)ªº¸ÑÄÀ¡G

(i) ¦pªG $x_0\leq \frac{1}{\eta}$¡A¥Ñ©ó x(t) ·|»¼´î¡A $x(t)\leq x(0)=x_0 \leq \frac{1}{\eta}$¡C¤S

\begin{displaymath}
y' = \alpha y \; (x-\frac{\beta}{\alpha})
= \alpha y(x-\frac{1}{\eta}) \leq 0
\end{displaymath}

¥ç§Y y ±q¤@¶}©l´N·|»¼´î¡C

(ii) ­Y $x_0>\frac{1}{\eta}$¡A

\begin{displaymath}
y'(0)=\alpha y(0)(x(0)-\frac{1}{\eta})>0
\end{displaymath}

©Ò¥H y(t) ¦b¶}©l®É´N·|»¼¼W¡C¦ý $x_{\infty}$ ¤£¥i¯à $ \geq \frac{1}{\eta}$¡]§Y x(t) ¤£¥i¯à«í $ \geq \frac{1}{\eta}$¡^¡A¦]¬°³o¼Ë¤@¨Ó y'(t) «í $\geq 0$¡A«hy(t) »¼¼W¡A $y_{\infty}\neq 0$¡A»P(1)¥Ù¬Þ¡C ´«¥y¸Ü»¡¡A¦pªG¦³®Ä±µÄ²¼Æ $\eta x(t)$ «í $\geq 1$¡A«h¶Ç¬VÂX´²¥Ã»·¤]°±¤£¤U¨Ó¡C©Ò¥H x(t) ²×¨s­n¤p©ó $\frac{1}{\eta}$¡A¦ý³o»ò¤@¨Ó¡Ay'(t) ¦b x(t) »¼´î¨ì $\frac{1}{\eta}$ «e¡Ay'(t) > 0¡F ·í x(t) ¶V¹L $\frac{1}{\eta}$ ©¹ $x_{\infty}$ »¼´î®É¡A y'(t)<0¡A¦]¦¹ y(t) ¥ý»¼¼W«á»¼´î¡A¥B¦b $x(t)=\frac{1}{\eta}$ ®É¡Ay(t) ¹F¨ì³Ì¤j­È¡C

¤U­±²ßÃD¬O¥Î¥t¤@­Ó¤èªk¡]¬Û¥­­±¦±½uªk¡^¨Ó¸ÑÄÀ(2)¡C

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°²³] $\eta >1$¡A§Q¥Î¤èµ{²Õªº«e¨â¦¡¡A»¡©ú

(1) $\frac{dy}{dx}=-1+\frac{1}{\eta x}$¡A¨Ã¸Ñ¥X $y = -x + \frac{1}{\eta}\ln x+C$¡C
(2) »¡©ú¸Ñ¦±½u¦b $x=\frac{1}{\eta}$ ¦³·¥¤j­È¥B¹Ï§Î¥W¦V¤U¡C
(3) ¦b xy-¥­­±¡Ax>0, y>0, x+y<1 ªº°Ï°ì¤¤¡A¥H C ¬°°Ñ¼Æ¡Aµe¥X¤@¨Ç¸Ñ¦±½u¡A¨Ã¼Ð©ú¦±½uÀHµÛ®É¶¡Åܤƪº¤è¦V¡C



µù¡G¥ú±q³o­Ó²ßÃD¬Ý¤£¥X¬°¤°»ò $y_{\infty}$ ¥²¶·µ¥©ó 0¡C

²ßÃD¡G
¥Ñ¤W²ßÃD»¡©ú $1-\frac{1}{\eta}<z_{\infty}<1$¡C
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¦pªG $\eta <1 $¡A¤ÀªR¾ã­Ó¶Ç¬V¯f¼Ò«¬¡]©w©Ê¤ÀªR©Î¤W²ßÃDªº¤èªk³£¸Õ¸Õ¬Ý¡^¡C

   
 
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¦b«e­±¨Ò¤l¤¤´£¨ìªì©l­È x0 = 0.1 ®É¡A§Ú­Ìı±o«Ü¯îÂÕ¡C¦ý¬O¦pªG±q¥t¤@­Ó¨¤«×·Q¡A¤]³\¨Ã¤£§¹¥þ¦p¦¹¡C¦pªG§Ú­Ì¯u¯à±N x0 ±±¨î¦b $ <\frac{1}{\eta}$ ªº½d³ò¤º¡A«h¤£ºÞ y0 ¬O¦h¤Ö¡A¤èµ{¦¡ $y'=\alpha y(x-\frac{1}{\eta})\leq 0$ «í¦¨¥ß¡C¦]¦¹¶Ç¬V¯f«K³Q¦³®Ä±±¨î¤F¡C

³o¥¿¬O¹w¨¾±µºØªº­«­n©Ê¡A¥Ñ©ó¬Ì­]ªºª`®g¡A¤@­ÓµL©è§Ü¤Oªº°·±d¤H¥i¥HÂಾ¦¨¨ã§K¬Ì¤Oªºªv¡ªÌ¡A¦b¤j³W¼Òªº¬Ì­]±µºØ«á¡Ax0 ¥i¥H¯u¥¿Åܤp¡A¦]¦¹¦³®Ä±±¨î¶Ç¬V¬Ì±¡ªºµ¦²¤¡A´N¬O§Q¥Î¹w¨¾±µºØ±N x0 ´î§C¨ì±µÄ²¼Æªº­Ë¼Æ¥H¤U¡]§Y¦³®Ä±µÄ²¼Æ <1¡^¡C

¨Ò.¡]Äò«e¨Ò¡^
°²³]¯u¯àµo®i¦¹¬y¦æ©Ê·P«_ªº¬Ì­]¡A­Y¹w¨¾±µºØªº¤ñ¨Ò°ª¹F

\begin{displaymath}
1-\frac{1}{5}=80 \%
\end{displaymath}

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