({ 11.2) ø. ½È_ (Comprtment Models) cqù¾ x 1 (t) D x 2 (t) 5Èí àçý. x 1 (t) D x 2 (t) Èí GÉ (dynmics) ªý Aà-í(4Í$: dx 1 dt = ( + c)x 1 + x 2 + I dx 2 dt = x 1 ( + d)x 2 w2 I,,, c, d ÑÝŠ. (1) I > 0 (ý ìíx@, input): Í$ (1) uøý 5Ÿ,, (4}j Í$ (inhomogeneous liner differentil equtions with constnt coefficients). à, lj ì 0í %e I v, Éä (soil, x 1 (t)) DTÓ (plnts, x 2 (t)) Èô}í. I = 0: Í$ (1) uø5ÿ,, (4}j Í$ (homogeneous liner differentil eqution 1 2 çí:pm
with constnt coefficients). Wà, c à/ø ¾í Ó( ( I = 0, ÄÑ.yJì Ó ¾), (x 1 (t)) D (x 2 (t)) 2 Óëí. ñ : «n I = 0 í8. íl, ø<òh, í! : (i) Ó./¼ÜòBø_Cù_½È. (ii) Ó.¼ÜòBBýø_½È}ÖÓ. üãòí«n à-: Í$ (1) óçkä³j w2ä³ A = dx dt = Ax(t) ( + c) ( + d) /,, c, d BýøÑ ( Ñø{,, CÌ í8 (trivil cse),?¹, Ó êr.¼ ). 2 2 çí:pm
íl, τ = tra = ( + + c + d) < 0 (ÄÑ,, c, d 0 /BýøÑ ). Ä, τ 2 4 = det A = ( + c)( + d) = d + c + cd 0 = ( + + c + d) 2 4( + c)( + d) + 4 = ( + c) 2 + 2( + c)( + d) + ( + d) 2 4( + c)( + d) + 4 = ( + c) 2 + ( + d) 2 2( + c)( + d) +4 = ( + c) ( + d) 2 + 4 0 I λ 1, λ 2 Ñ A íôm, ;W,H )í τ,, J τ 2 4, ªRû (1) τ 2 4 0 λ 1, λ 2 Ñõ. 3 2 çí:pm
(2) 0 λ 1, λ 2 UCBýøÑ 0. (3) τ < 0 λ 1, λ 2 ÑŠ ( > 0), CøŠ, ø 0 ( = 0). ;W,Hí!, Ê τ 2 4 > 0 -, } «ns8 à-. 8 1. > 0: (0, 0) Ññøí õ/uø ì õ (stle node, C ü, sink). <2: ÄÑ = d + c + cd > 0, (i), d > 0 àçý ¼m (ÄÑÊù_½È øj ìàp 0¼ Ó í ); C (ii), c > 0 àçý ¼m (ÄÑÊø_½È øj ìàp 0¼ Ó í ); C (iii) c, d > 0 àçý ¼m (ÄÑÊø_D ù_½èì} Aí, J ìàp 0¼ Ó í ). 4 2 çí:pm
ã,híúª?, Ìø_Ëû õ (0, 0) Ñø ìõ ( ü ). 8 2. = 0: Î7 (0, 0), w í õ. <2: ÄÑ = d + c + cd = 0, â ªû d = 0 / c = 0 / cd = 0 (i) = c = 0 àçý -Ê½È 1 (ÄÑø _½È³¼ Ó í ); C (ii) = d = 0 àçý -Ê½È 2 (ÄÑù _½È³¼ Ó í ); C (iii) c = d = 0 àçý ³Ó ¼ Í$, ¹,Ó \MìM,.Z (ÄÑøDù_½ÈÌ Ì¼ Ó í, Ó cêí$q¼ ). ÇÕ, víí$ GÑ dx 1 dt dx 2 dt = x 1 + x 2 = x 1 x 2 5 2 çí:pm
ø,ùó, ªû d(x 1 + x 2 ) dt = 0 Ä, úfí t 0, x 1 (t) + x 2 (t) = 6ÿuz,,Ó \MìM,.Z. W. tj c = d = 0 í8. <j> íl, ä³ A = â ª) J = 0 τ = ( + ) < 0 Ä, ùôm} Ñ λ 1 < 0 / λ 2 = 0 6 2 çí:pm
y;w τ = λ 1 + λ 2, ) λ 1 = τ = ( + ) QO, } ÔMFú@íÔ²¾à-: (i) λ 1 = ( + ) ú@íô²¾ u = Ûjä³j u1 u 2 Ç(, óçkj(4j Í$ = ( + ) u1 u 2 u1 u 2 : u 1 + u 2 = u 1 u 1 u 1 u 2 = u 2 u 2 cü(,?óçkj u 2 = u 1 u 1 = u 2 ÄÑ, 2BýøÑ, )gíj FJ, ª Ô²¾ u 1 = u 2 u = 1 1 7 2 çí:pm
(ii) λ 2 = 0 ú@íô²¾ v = çkjä³j v1 v 2 Ç(, óçkj(4j Í$ 4gk Ä, ª Ô²¾ v 1 + v 2 = 0 v 1 v 2 = 0 v 1 + v 2 = 0 v = v1 v 2 = 0 v1 v 2 : 4ó (iii) ;W½LŸÜ, øoj x(t) = c 1 e (+)t 1 1 = c 1 e (+)t 1 1 + c 2 e 0 t + c 2 ÄÑ + > 0, Í$íÅ WÑ lim t x(t) = c 1e 1 + c 1 2 = c 2 8 2 çí:pm
½. c 2 ÑS?. Ê t = 0 v, x1 (0) x 2 (0) = c 1 1 1 + c 2 4óçk c 1 + c 2 = x 1 (0) c 1 + c 2 = x 2 (0) ø,ùó, ) c 2 ( + ) = x 1 (0) + x 2 (0) Ä, w2 c 2 = x 1(0) + x 2 (0) + = K + K def = x 1 (0) + x 2 (0) = x 1 (t) + x 2 (t) úfí t 0. Ä, / lim t x 1(t) = lim t x 2(t) = K + = K K + = K + + 9 2 çí:pm
,H! íø_ Üj : âà-íçý ) x 1 x 1 x x 2 2 (i) Ó ªpø_½Èíóú 0Ñ lim t x 1(t) = K + (ii) Ó ªpù_½Èíóú 0Ñ lim t x 2(t) = K + +, +, ð : K + K + + + AK üõñø õ, ÄÑ = K = K = K 0 0 + + + = 0 + + 10 2 çí:pm
W 1. t Hà- G x 1 0.1x 1 /hr 0.5x 2 /hr 0.2x 1 /hr x 2 í}j Í$1}& õ (0, 0) í ì4. <j> (i) ;WÜ: 0 = ¼pí 0 ¼ í 0 ªû H Gí}j Í$Ñ dx 1 dt dx 2 dt = (0.2 + 0.1)x 1 + 0.5x 2 = 0.2x 1 0.5x 2?óçkä³j dx dt = 0.3 0.5 0.2 0.5 x(t) def = Ax(t) (ii) }& õí ì4: = det A = 0.15 0.1 = 0.05 > 0 11 2 çí:pm
/ J τ = tra = 0.3 + ( 0.5) = 0.8 < 0 τ 2 4 = 0.64 0.2 = 0.44 > 0 FJ, (0, 0) uñøí õ/ùóæôm λ 1 < 0, λ 2 < 0, 4ÄÑ τ 2 4 > 0 ýômñùóæ õ, > 0 û U, J τ < 0 ) ÑŠ., õ (0, 0) Ñø ìõ (stle node, ü, sink). 9õ,, õòlôm, ª) 0.3 λ 0.5 det 0.2 0.5 λ J = (0.3 + λ)(0.5 + λ) 0.1 = λ 2 + 0.8λ + 0.05 = 0 λ 1,2 = 0.8 ±.64.2 2 ÌÑŠ. = 0.4 ± 1 2.44 W 2. ààp ¾í Ó(, I x 1 (t) Ñ 2 Ó íë/ x 2 (t) Ñ 2 ÓíëJ á KÑ x 1 (0) = K, x 2 (0) = 0. 12 2 çí:pm
D È Óëí Ñ àçý, dx 1 dt dx 2 dt = x 1 (t) (2) = x 1 (t) (3) x 1 x 1 x 2 t (4Í$íj. <jø> â (2) ø, x 1 (t) = C 1 e t H M x 1 (0) = K, ) K = x 1 (0) = C 1 e 0 = C 1 FJ, x 1 (t) = Ke t QO, ø,hp (3), ) dx 2 dt = Ke t 13 2 çí:pm
FJ, x 2 (t) = Ke t dt H M x 2 (0) = 0, ) Ä, / (, ã J,F), ªû = Ke t + C 2 0 = x 2 (0) = K + C 2 C 2 = K x 2 (t) = K(1 e t ) x(t) = Ke t K(1 e t ) J Í$íÅ Æ Ñ 0 lim t x(t) = K 0 4ø Üí!, / ÑÇø_ õ. K <jù> íl, Í$íä³ 0 A = 0 14 2 çí:pm
â ) = det A = 0 0 = 0 FJ, Î7 (0, 0) Õ, í õ. / τ = + 0 = τ 2 4 = 2 > 0 FJ, ùôm λ 1, λ 2 ÑøŠ, ø 0. yxñk, â τ = ÑùÔM5, ª) λ 1 = / λ 2 = 0 C6âÔMDÔ²¾íì2 5, à-. (i) ÔM: 4óçkj det(a λi) = λ 0 det λ = ( λ)( λ) = λ(λ + ) = 0 FJ, λ 1 = J λ 2 = 0 15 2 çí:pm
(ii) Ô²¾: λ 1 = ú@íô²¾ óçk u U) 0 0 u = u1?óçkj(4j Í$ u 2 u1 u 2 = u1 u 2 ÄÑ 0, â,ª) FJ, ª u 1 = u 1 u 1 = u 2 u 1 = L<Ý 0 ím u 2 = u 1 u = 1 1 QO, λ 2 = 0 Fú@íÔ²¾ 4óçk v U) 0 0 v = v1 v1 v 2 v 2 = 0 v1 v 2 16 2 çí:pm
?óçkj(4j Í$ v 1 = 0 v 1 = 0 ÄÑ 0, â,) v 1 = 0 v 2 = L<Ý 0 í FJ, ª v = 0 1 (iii) ;W½LŸÜ, Í$íøOj x(t) = c 1 e t 1 + c 1 2 e 0 t 0 1 H M ) K 0 = = c 1 e t c 1 e t + c 2 x(0) = K 0 c 1 e 0 c 1 e 0 + c 2 = c 1 c 1 + c 2 17 2 çí:pm
â û c 1 = K c 2 = K Ä, Í$íÔyj x(t) = Ke t K(1 e t ) Å. Ñ *0 (excretion rte). íj à-: â ª) si AÍú ln, ) FJ, â eú x 2 = K(1 e t ) K x 2 (t) = Ke t ln(k x 2 (t)) = ln K t (t, ln(k x 2 (t))) ª)é0Ñ íò(. Ä, = ( ò(íé0) ( 1) 18 2 çí:pm