# Y.Y.Tseng Functional Analysis Research Center, Harbin Normal University, Harbin, Heilongjiang, , P.R.China. Department of Mathematics, College

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1 ³ ¹µ± ² ÄÐÝ Y.Y.Tseng Functional Analysis Research Center, Harbin Normal University, Harbin, Heilongjiang, , P.R.China. Department of Mathematics, College of William and Mary, Williamsburg, Virginia, , USA.

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3 Ì Ä ¹ ÐÍ Æ ½ ÐÍÐ Æ ÐÍÍ º ¼ ÓÞÆ ³Î ¼ ÓÞ Í ÆÎ ÐÍ Ì ¹ ÐÍ ÜÇ ÐÍ ÜÇ ¼ ÓÞ À Î Æ ³Ì (nondimensionalization) ÜÇ Æ Õ ½ »Î Í ÌÊ Í Æ» » º ¼ Ô º ß Î ¾ Æ» ¹Í Ì Í ÈÓÄ Banach» Ý Ý» Æ» Æ

4 1 4.5 Æ»ÆÐÎ ÞÆ Turing µ ÕÍ ÌÙ ¾ Æ Ô Û Æ Ò¼ Æ Ô ½Ë »Í Ì Ó ¼ Hamilton » ± Æ» ÞÆ ³ Ë Ò¼ Æ Ô 85

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6 Í Å º 1.1 ÓÕÑ Ê ÕÞ ÉÐÐÌ¼Þ Ð ³ Æ Æ ÓÎ Í Ä ¹ÒÓ (first order scalar ordinary differential equation), ÞÅ (autonomous equation)± Ç y : S R, S R ÍÐ Ý y 0 Í ÆÔ (initial value) (1.1) Ô (A) Ý (analytic method) ½ Ð Æ Æ Ç dy dt = f(y), y(0) = y 0, (1.1) dy dt = ky, y(0) = y 0 Æ y(t) = y 0 e kt ; dy dt = ky(1 y), y(0) = y y 0 e kt 0 Æ y(t) = 1 y 0 + y 0 e kt. Í f(y) Ý Ð Ð (B) (numerical method): Ç Æ (1.1) Í dy dt y(t + t) y(t) t y(t + t) = y(t) + f ( y(t) ) t (1.2) y(0) = y 0 (1.2) ³ Ü² ÜÐ Ð ßº Euler Í Ô Æ 1.3 Ï «Ã (C) (Qualitative Method) ½ ÆÉ Ð ÆÜ Å Æ Ü (equilibrium, steady state) Í (1.1) y(t) = y 0 Æ Þ f(y 0 ) = 0. Á (phase line) Í (1.1) ÆÃ ÆÐÐÒ É 1.1. y = y(1 y) 2 (y 3), y(0) = y 0. (1.3) ÃÉ µ y = 0, y = 1 Å y = 3. Ã (asymptotic behavior) Í º Æ Í t, ß Í y(t). 3

7 y t 1.1 ( ) (1.3) ÂÀ, (µ) (1.3) È 1.2. (a) y = y(1 y) 2 (y 3), y(0) = 2 (1.4) È Ý ÅÐ Ñ Æ, Òµ (, ), Ë lim y(t) = 1, lim t Đ²ÆÍ ÃÉ (converges to an equilibrium). y(t) = 3, t (b) y = y, y(0) = 1 (1.5) Ð y(t) = e t, Æ Òµ (, ), lim y(t) = 0, lim y(t) =, t t»î (tends to infinite in infinite time). (c) y = y 2, y(0) = 1 (1.6) Ð y = 1 1 t, Æ Òµ (, 1), «t 1+ Ù y(t), º Ö Ù Ù Ä (finite time blow up). (d) Ð y(t) = 2 1 2t, Òµ (, 1 2 ), y = 1, y(0) = 1 (1.7) 2 y lim y(t) =, lim t y(t) = 2, lim y (t) =, t (1/2) + t (1/2) Ù Ù Ä (derivative blows up in finite time) µù (1.1) 1. f(y) Ù y = y 0 ¾ ±Ã Ú Ù ε > 0 Ü (1.1) Ù ( ε, ε) Í

8 y y y y t t t 0 1/2 t 1.2 Ö Ä Û (1.4) (1.5) (1.6) (1.7) È 2. f(y) Ù y = y 0 ¾ ±Ã ³ Ú Ù ε > 0 Ü (1.1) Ù ( ε, ε) ÍÍ 3. y(t) È ÆÍ Ù I = ( T 1, T 2 ), 0 < T 1, T f(y) Ù I Ñ±Ã ³ Ú y(t) Ù I Ñ µæ ÛÂ µæ ÙÂ µíãé 5. f(y) Ù I Ñ±Ã ³ Ú y(t) Ù I Ü µ µ a) Í ÃÉ b)»î c) Ä d) Ä (1.1) Á Æ 1. ÐÐ ÐÜ ÖÒÆÒ ; 2. Ð f(y) Æ½ «¾ Æ ; 3. ½ «¾ ÆÁ f(y) ÆÁĐ ÐÁÆ f(y) < 0, ( ); f(y) > 0, Ã ( ). ÍÁ Æ : y 0 Í (1.1) ÆÐ Ü ß δ > 0 Å Æ y (y 0 δ, y 0 + δ), y ÔÆ lim t y(t) = y 0, ÃÜ y 0 Íµ Æ, Ð (sink); ½ ß δ > 0 Å Æ y (y 0 δ, y 0 + δ), y ÔÆ lim t y(t) = y 0, ÃÜ y 0 Í µ Æ Ð (source); Í Í ÆÜ Ï (node). Ð Ü Æ ³ ÜÁ Ü ±ÈÆÁÆ É Ñµ Í Þ Ä Þ ÑµÜ ÅÜÞ ÅÁ É Ø ÇÆ Í± Æ

9 y y y Þ ( ) (ß) (µ) 1.2. f(y 0 ) = 0, f(y) ÙÍ (y 0 δ, y 0 + δ) Ñ±Ã ³ (1) f (y 0 ) < 0, Ú y 0 ÐÍ (2) f (y 0 ) > 0, Ú y 0 ÐÍ Ô (3) f (y 0 ) = 0, º»² ÃÉ 1.2» ÓÕÑ ÊÕÆ Ð ÆÃ ÞÅ µ Æ Ä Ñ ÓÄ ( Ó Ó, Ó )(bifurcation) µù y = y(1 y) h, (1.8) Ùß y(t) Í ½½ h ½Ù Ù (1.8) Ã É Æ y = 1 ± 1 4h. 2 0 h t t 1.4 (1.8) È ( ) (ß) h < 1 4, (µ) h > 1 4 h = 1 4 Ð (1.8) È «h < 1 4 Ù (1.8) ³ ÃÉ «h = 1 4 Ù³ ÃÉ Ç µí «h > 1 4 Ù (1.8) ÃÉ ÛÑ (1.8) ÃÉ h = y(1 y) h = y(1 y) Ù h y Ñ µðí ÓÄÐ (bifurcation diagram). È Ì ÐÍ Ç ß º µ Ã h Ð Ú 1.4 Å º y(0) = 1 2 µü y h(t), «h < 1 4 Ù y h(t) Đ²ÆÍ ÃÉ «h > 1 4 Ù y h (t) Ù Ù µ 0 ÛÑ Ù Ù ÄÆ Å. Ð Ò Ð ÛÅ Æ y = f(λ, y), (1.9)

10 ÅØ ÇÆ Å (λ, y), λ Æ y ÆÜ ², (1.8), Ü f(λ, y) = 0 f y (λ, y) = 0 (1.10) y(1 y) h = 0 1 2y = 0 (h, y) = ( 1 4, 1 2 ). Ü 1.4 ĐÅ Í Ò¼ f(y) = y(1 y) h» h Á Æ ÅÇ Ä Ã É Ð ÇÆ Ä Ð Æ «º Ä Í ÕÐ Ð»Æ 1.4. (a) 1.3 µù (1.8) ÈÖµ» ÏÓÄ (saddle-node bifurcation). Ù È Í Ð ³ ÃÉ ¼ÍÐ ß È Ù È ¾ ÐÀ½ ÍÍ µù (1.9), Ë È± ÛÐ È µ (λ 0, y 0 )Å(1.10) ³ µùå (b) µù 2 f λ 2 (λ 0, y 0 ) 0, 2 f y 2 (λ 0, y 0 ) 0. (1.11) y = y(y 2 λ). (1.12) È Ý λ = 0 ÐÍ È Ù È ÍÐ ÃÉ ¼ÍÐµÍ ß ÈÖµ ÓÄ (pitchfork bifurcation). Bifurcation Í Ô Ñ bifurca, Ð ³ Ò Ñ Ñ È Ð bifurcation Ó ± ÛÐ (1.10) Å f λ (λ 0, y 0 ) = 0, 2 f λ y (λ 0, y 0 ) 0, Æ f λ (λ 0, y 0 ) = 0 ß Û²² f(λ, y 0 ) 0 (c) µù 2 f y 2 (λ 0, y 0 ) = 0, 7 3 f y 3 (λ 0, y 0 ) 0, (1.13) y = y(y λ). (1.14) λ = 0 ÐÍ È È ³Ð Ð³ ÃÉ ÅÃÉ 1.2, y = 0 Ù λ < 0 Ù Ù λ > 0 Ù ÃÉ λ < 0 Ù y = λ Æ λ > 0 Ù y = 0. ß ÈÖµ ÓÄ (transcritical bifurcation), ± Ûµ (1.10) Å f λ (λ 0, y 0 ) = 0, 2 f λ y (λ 0, y 0 ) 0, Å Ñ È ÍÍ λ, y = y 0 ÐÍ Ô (d) µù 2 f y 2 (λ 0, y 0 ) 0. (1.15) y = y(y 2 ε) λ f(ε, λ, y). (1.16) «ε = 0 Ù (1.16) µ y = y 3 λ, Æ ÍÃÉ y = 3 λ Ù λ = 0 ÙµÍ Ç f y (0, 0, 0) = 2 f y 2 (0, 0, 0) = 0, 3 f (0, 0, 0) 0. y3

11 8 1.5 ( ) ( µ) Ë ( ) ( µ) ÙÍ ε > 0 (1.16) Ù λ Ù ÃÉ Æ È Ñ ³ µ Ë ßÍ ÈÛÐÑÐÍ ³ È Ù ε = 0 Ù Ç ÖµÍ ÅÏ (cusp). È Ö È± ÛÐ (1.10) Å 1.3 ÕÑ ÃÕÆ 1.6 Ê ( ) ε < 0, (ß) ε = 0, (µ) ε > 0 f λ 2 (λ 0, y 0 ) 0, 2 f y 2 (λ 0, y 0 ) = 0, 3 f y 3 (λ 0, y 0 ) 0. (1.17) 1.1 ÏÞ Ä ĐÅ (1.1)» Ð (1.2) ÆÍ Ð Ä Ð y n+1 = f(y n ), y 0 = a (1.18) Æ Ó (difference equation) (map). a R, f C 1 (R). {y n : n = 0, 1, 2, } ÍÐ Á (sequence). Ç ÐÆÍ º Í ÄÕÐ y n+1 = ay n + b, y 0 = c Æ y n = y + a n 1 (y 0 y ), Þ y = b Í ÆÜ a 1. a = 1, Ã Ä 1 a Å Í y n = y 0 + (n 1)a. a 1 (y n y ) ÍÐ ½

12 9 Í (1.18), Ü y y = f(y). (1.19) Ø ÄÐ Ü (fixed point). Å (1.18) Ð Ü y Å Æ ½ Ü³ ÐÐ Æµ 1.3. f C 1 (R, R), f(y ) = 0, (1) f (y ) < 1, Ú y Ð (2) f (y ) > 1, Ú y Ð y n+1 = y + f (y )(y n y ), (1.20) f (y ) < 1 y 0 y ÆÐ Î¼¾ lim n y n = y. Ø «(1.1), {y n } 0 > f (y ) > 1 ÍÐ y Æ ½ {y n } 0 < f (y ) < 1 ÍÐ y Æ y y y * y * n n 1.7 ² Ä ( ) 0 > f (y ) > 1, (µ) 0 < f (y ) < µù y n+1 = ry n (1 y n ), y 0 = a. (1.21) Þµ y = 0 ÐÍ Ã³ y r = 1 1 r ÐÍ r Æ f (y r ) = r + 2, Ú r = 1 ÐÍ Ö Ü f (y r ) = 1, «r < 1 Ù y r µ¼ «r > 1 Ù y r µ «1 < r < 3 Ù y r ÐÍ «r > 3 Ù y r ØÐ ßÙË Æ (1.21) Ý Å È (i) «3 < r < 1 + 6(Õ 3.45) Ù Æ Í Åµ 2 (ii) «r Ù 3.45 Å 3.54 Ù Æ Í Åµ 4 (iii) «r Í ÛÔ Åµ 8, 16, 32 ÑÝ ßÍ Öµ Á¼ (period doubling cascade); (iv) «r ÛÅ 3.57 Ø ĐÉ (chaos), Å Ü µ Ü ±Ã ÛÑßÙË (1.21) Åµ 2 n Å Ë Åµ Æ n Å (v) «r Ù 3.57 Å 4 r (1.21) Ñ Ð ÅÌ Í Ð Ð Đ²

13 10 r>4 (vi) ZT " %l/ 9 r = 1 Æ 4 +P T g F y 1.8 ypl_ + (#) } 2 [ ({) } 4 [ ()) ) x lfsf Æ 7V?>XKBÆ a,l\ FE *KB : \ }<Q D \$<) ^ 6 U,L q l E ÆB U D5?Æ x }Y) F }Y) Z ÆB ^ gjol O+, B<,L 9 x ) F \ % } B x Y } 3 nu sæ 2,L q Z Æ ol l ÆZU Æ F U B< Æ Z F } ` } F9 Z sæ D Æ Z b } I B l Z \ B ^ F * U Æ Zz }Y I07oL E qe..y, O B 5?!,L q l B< U B< Æ : E z *3 K =e< 1.9 Logistic 1.5 8C # 1 < r < 4; ( Logistic r=3 (flip bifurcation). Hopf f (yr ) = 1 (Hopf bifurcation). 2 2 y 3 < r < 4. 2 f = f f (f 2 ) (y ) = 1 (period doubling bifurcation). 2 3 < r < 3.57 (1.1) (1.2) yn+1 = yn + f (yn ). = ^U t = 1, F F (1.1) (1.22) \Y_O/ (1.18) Æ_OB ZX) 1.3 zæbjpu f (y ) = 1 1 UO 1.2 zæ f (y ) = 0 \ Y Q % (1.22) B< 1 (1.18) 7 U} % Z Æ x 1.4 ^ \x` DRN82? 3 y(t) }Y?0 } Æg' t = D E \G! F Æ } V UMalthus = (Malthus, 1798) y = ky, k > 0. (1.23)

14 11 Malthus Î Æ y(t) = y(0)e kt ÉµÐ ³ÆÕ Ç Ð ³ĐÆ ÆÐ Û Ø Ç ÞÐ ÇĐ½ Î Ø Æ Í y = yf(y), (1.24) f(y) Í Ç (grow rate per capita) y Ç y ÍÐ ³ ½ f(y) = y y 1838 Vehulst ÐµLogistic y = ky(1 y ), (1.25) N k > 0 Đ Ç N > 0 Í (carrying capacity) Ä 1.3 Þ ÆÍ ÆÆ Logistic Î y = ky(1 y ) h, (1.26) N h Í (1.25) (1.26) Æ 1.1 Ï 1.2 Ï µ (1.26) ÞÆ h = kn 4 º Đ (Maximum Sustainable Yield) ½ h > h Ð y(t) Ð ± ÊÝ (1.26) Þ ³ Ð ÞÍ Æ Ó Ð Ê ÝÆÐ Æ Ð Ô¼ ¾ÐÐ (Logistic equation with constant harvesting effect): y = ky(1 y ) Ey, (1.27) N E Æ ½ ³ Ey Í Ð ³ ½Æ k > E (1.27) ÆÁ (1.25) ÐÝ ½ k < E Ð ÊÝ ¾Ð ÀÎ ÍÙ±Allee ÆÐ Î : y = ky(1 y N )( y M 1), (1.28) 0 < M < N Ä² N ½ M (sparsity constant) (1.28) Ù±± µ Ü y(t) = 0 y(t) N ½ y(t) = M Í µ Æ ÍÙ± µ (bistability) ÆÎ 1.10 ( ) (µ) º ÆÐ ĐÎÆ Í ¼Ó Æ (1.26) (1.27) Í Î Æ± Ç Æ ³ÍÐ r > 0 º Æ ³ y = g(y) rc(y), (1.29) g(y) Í ½ ÆÇ ½ c(y) ÍÁ Æ ³ g(y) Æ± ½Í (a) Logistic Ç (b) Allee Ï Ç ½ c(y) Ä Đ ĐÓ Holling Æ Ý (predator functional response) ÆÉ (1.26) (1.27) ÞÆ Ý Í Ý c(y) = h Ý c(y) = Ey Holling Ð¹Ð Æ Ý

15 12 ( I) Ey, 0 < y y, c(y) = Ey, y y, (1.30) ( II) ( III) c(y) = c(y) = Ay, y > 0, (1.31) B + y Ayp B p + yp, y > 0, p > 1, (1.32) A, B ¹ Ý Æ «ÜÍ (C1) c(0) = 0; c(y) Í ÇÝ (C2) lim y c(y) = c > À Þ ( ) À ( µ) ± À ( ) Holling I; ( ß) Holling II ( µ) Holling III. ÐÐ Holling II Æ Ý Í Ivlev c(y) = A Be ry. (1.33) Í Ivlev» Holling II ÇÆ Ü «Á«Ý ÐÆ² «Í Î Æ Ó Ü ĐÞÆÍ Ý Æ Ü Í Ç Ø c(y) (C1) (C2) (C3) c (0) > 0 c (y) 0 y 0 «ÔÔ Ã Ä c(y) ÍÑ«Æ Holling II ½ c(y) (C1) (C2)

16 13 (C4) c (0) = 0 ß y Å c (y)(y y ) 0 y 0 «ÔÔ Ä c(y) ÍÑ«Æ Holling III Ð «Holling I,II Ivlev «Ñ«Æ Holling II (1.29) (Noy-Meir,1975; Ludwig et.al.1978; May, 1977) Þº± ÐÐ» ĐÎ (Noy-Meir,1975) Þ y(t) ÍÐÐÓ¼ Æ ³ ½ r Í Ç ¼Æ ³ (Ludwig et.al.1978) Þ y(t) ÍÐÐ¼ºÚ (spruce budworm) Æ ³ Ä 1.2 ÏÞÆ (1.29) ± ½ g(y) ÐÐÑ«Æ Logistic Ç ½ g(y) = rf(y), f(y) f(0) > 0, (0, N) f(y) f(n) = 0 ( Ä y(t) > N Æ ) c(y) ÍÑ«Æ Holling II Ý Ä r Å Ã r = g(y) c(y) É Æ 1.12 : g(y) Ò Logistic È, c(y) ÎÒ Holling II. r = f (0)/c (0) Î y = 0. ¾ y = 0 Î Þ. Å Þ µ ± Ç c(y) ÍÑ«Æ Holling III Ý (1.29) : g(y) Ò Logistic È, c(y) ÎÒ Holling III Þ c (0) = 0 y = 0 ÍÐ µ ÆÜ ½ ÝÆ g(y), c(y) µ ²Ó Ð ( ) ±± Noy-Meir, May Æ ³Þ ± ÆÇ ¼Ë (r ½Ë ) (1.29) Ðµ Ü Î N Ó¼ ³ º Ç ¼Çº (r Ç ) ÃÓ¼ ³ Æ ¾Ðµ Ð Ó¼ ³² ³ ĐÆµ Ü ÐÜ Í r ÐÁÇĐ Øµ Æ r = r, ĐÆµ Ü Þ Í

17 14 Ó¼ ³ ÇØÎÆµ Ü Ó¼ ³Æ¼Û ÇÅ ßÆÇ ¼ß ¼ ² r Ç ½ r Æ Î Ç ÅÇ ¼ ³ Îµ Ü Æ ¾ Ó¼ ³ ³ÐÜ ÒØÇ ¼ r Æ ³ÇØÎ ¾Ð r = r Ü½ Ð ÝÆ Ô ØÜÆ Æ Ð ½ÃÆÆØº Đ ÞÐ ¼Ó¼Æ º Hysteresis Ä ÅÇ ± HysteresisÆ Ë 1.14 Hysteresis ÌÊ (1.29) ßÙ±Ð (??). º HysteresisÆ Í Æ ÅØ Î Æ. Hysteresis Ë Ó Þ Æ Đ Ð µ Ü Đ Å¾Ðµ Ü ĐÆØ ½ Í Æ ¾Ð Ð Ð ËÆ ¼Î y = a by + ryp h p + yp, a, b, r, h > 0. (1.34) a, b, h > 0 r Å (1.34) Æ ß Ð Hysteresis Æ Đ (1.34) (Carpenter et.al.1999, Scheffer et.al. 2001) Þ Ð y(t) ÍÐ ÞÐ Ü¼Æ ³ a Í Ü¼Ç b Í Ü¼º r Í Ü ÂÔ x r 1.15 (1.34) a = 0.5, b = 1

18 15 ¼ÓÞ Î ÄÅ Ð ½Î Þ ÆÆ Ç ĐÞ ½ ÆÆ Ô Ü Ý Æ»ÜÜ ÐÓÄ Ä Ð ÇÆÎ (1.24) Ä «(1) f(y) [0, ) Í Æ (1.24) Logistic (2) f(y) [0, ) Í ÇÅ Æ (1.24) Allee Ï (3) f(y) [0, ) Í Ç Å Æ (1.24) Hysteresis Ð «Ç f(y) Æ ½ f(y) ½ Æ ÄÐ ß N Å y > N f(y) < 0 Allee Ï Hysteresis f(y) ±º ½ Ä «(1.24) Å Allee Ï Hysteresis ÇÆ «{y > 0 : f(y) < 0} ±± Å Ñ Ã Ä Ñ Allee Ï Hysteresis ( 1.16) (1.24), Allee Ï Hysteresis Logistic À±»Ü Ä ÅÇ Æ» 1.16 ±È f(y) ( ):Logistic ; ( ß): Allee ( µ): «hysteresis ;( ): «Allee ( µ): «hysteresis 1.5 Ü Æ» ÕÑ Ê º ¼Ó»ÆÍ É ¼Ð ÆÇ Ó ºÍ»Á Æ (») Malthus Î y n+1 = y n + by n dy n = (1 + b d)y n = λy n, (1.35) b Í Ð d Í Ô λ Í ß (per capita survival rate) (1.35) Æ Í y n+1 = y 0 λ n. λ > 1 Ð Õ Ç 0 < λ < 1 Õ Î y n+1 = y n S(y n ), (1.36)

19 16 S(y) Í ß ¼Ó ½Ë Ø Î Ðµ Ð Æ S(y) (Bellows,1981) Þ ÐÆ Bellows Î λy n y n+1 =, a > 0, b 1, λ > 0. (1.37) 1 + (ay n ) b (Hassell,1981) Þ ÐÆ Hassell Î y n+1 = (Ricker,1954) Þ ÐÆ Ricker Î (Beverton-Holt,1957) Þ ÐÆ Beverton Holt Î λy n, a > 0, b > 0, λ > 0. (1.38) (1 + ay n ) b y n+1 = by n e yn, (1.39) y n+1 = ry n 1 + y n. (1.40) ½ Ä 1.5 Þ Æ Logistic Í Í (May,1977) Þ ÐÆ ßÍ É ÐÚº Ð Äß Æ S(y) Ð (1) S(y) Í Æ Compensatory (2) S(y) Í ÇÅ Æ Allee Ï Compensatory Þ F(y) = ys(y) (1a) F(y) Í Æ lim F(y) = 0, under compensatory y (1b) F(y) Í ÇÅ Æ lim F(y) = 0, over compensatory y (1c) F(y) Í ÇÆ lim F(y) = C > 0, exact compensatory ( 1.17) y Bellows Î (1.37) Þ 0 < b < 1 under-compensatory b > 1 over-compensatory ½ b = 1 exact-compensatory. ÇÆ F(y) Ä± 1.4. F(y) = ys(y) Ð ÛÂ (1.36) Æ {y n } µ Û Ù Đ²ÆÍ ±ÐÆ ± Ä Å F(y) = λy(1 y)(logistic), F(y) = bye y (Ricker), F(y) = λy (b > 1) overcompensatory. 1.3 ÏÞ ÔÎ ½, Å«ÝÆÜµ Å 1 + yb Ç Ò Ø Æ.

20 Þ F(y).( ):under-compensatory ; (ß):exact-compensatory ;(µ):over-compensatory. 1.6 Î»Ð 1. Ð y = (y + 2)(y 5) y 4 ÆÁ Ü y(0) = 0, y(0) = 3.5 Æ ÆÃ 2. Ð y = y3, y(0) = 2 Æ Æ Đß y 1 3. Ð y = y(1 y) y 3y + a(1 y) Æ Ð 4. Ð y = y(1 y) 2 + k Æ Ð 5. Ð y = y(1 y 50 ) ay 5 + y Æ Ð 6. Beverton-Holt Î y n+1 = ry n 1 + y n Þ x n = 1/y n Ã x n Ð Í ÅØ Å Beverton-Holt Î Ç 7. Bellows Î y n+1 = λy n 1 + yn b (b > 2), a) Ð» ( Ü ) Ü b) Matlab Ð «λ ÔÆ Õ Ð 8. Ë Ð Î (1.29) y = g(y) rc(y), g(y) Ñ«Æ Logistic Ç c(y) Ñ«Æ Holling II Â (C1), (C2), (C3) (1.29) Æ Â 1.12 ÞÆ±Ð Þ Ð g(y), c(y) Æ Â Ë Þ Ý Æ Â

21 18

22 Í º 2.1 À Đ x = ax + by y = cx + dy Ù±± ³ x(t), y(t), a, b, c, d R. Ö É³ Ç ( ) ( ) ( ) d x a b x = dt y c d y dy dt = A Y, Y (0) = Y 0, (2.1) ( ) ( ) u a b Y =, A =. (2.1) Ù±Õ ÇÆ Y (t) = e λt V, ÆØ Ç µ (2.1) Ä Å v c d AV = λv, ½ (λ, V ) Ö A Æ Ô É³ Ø º ÐÁÆ ÐÂÅ Å λ 2 (a + d)λ + ad bc = 0, (2.2) T = a + d Í A Æ (trace), D = ad bc Í A Æ (determinant), (2.2) Đ A Æ Ô (2.1) ÆÐ Ç λ 2 Tλ + D = 0. (2.3) (a) λ 1 λ 2, Ã Y (t) = c 1 e λ1t V 1 + c 2 e λ2t V 2 ; (b) λ 1 = λ 2, Ã Y (t) = c 1 e λ1t V 1 + c 2 e λ2t (V 1 t + V 2 ), V 1, V 2 AV 1 λ 1 V 1 = 0, AV 2 λ 2 V 2 = V 1. λ 1 λ 2 Æ «² λ 1, λ 2 = α±iβ Æ Á Æ Ç Ä Æ ½ ÐÍ Æ ± Ð É Ç Í ÈÐ«(solution curve), ½ t-xy ÜÇ Æ Ç ( t, x(t) ), ( t, y(t) ). y(t) 0 x(t) 0 x(t) y(t) 0 x(t) y(t) t t t 2.1 Î À È ( ) ¹Þ, (ß) ¹Þ (µ) Ð 19

23 Î À Â ( ) ¹Þ, (ß) ¹Þ (µ) Ð ¾ ( x(t), y(t) ) Đ x y ÜÇ Ð ÆÕ x y ÜÇ ±Õ Æ ÆÅÐ (phase portrait), x y ÜÇ Å (phase plane). Æ Ç Á ÆÉ Ï Æ Ý ºÍ ½Í ÆÅ «(phase space)íõ Á Æ Đ Ö A Æ (2.3) Ü Ð (T, D) ÆÔ Ü Á λ 1 + λ 2 = T, λ 1 λ 2 = D, ³ Å (A) T > 0, D > 0, T 2 4D > 0, Ã λ 1 > λ 2 > 0; (B) T < 0, D > 0, T 2 4D > 0, Ã λ 1 < λ 2 < 0; (C) D < 0, Ã λ 1 < 0 < λ 2 ; (D) T > 0, D > 0, T 2 4D < 0, Ã λ 1, λ 2 = α ± iβ, α > 0; (E) T < 0, D > 0, T 2 4D < 0, Ã λ 1, λ 2 = α ± iβ, α < 0. Ð T-D Ð (Trace-Determinant plane) É D D B E D A stable unstable C C T unstable unstable T 2.3 µ - ÈÞÈ Ç» À T-D ÜÇ Ð Þ Æ T = 0 (A» C, B» C Ñ ), D = 0 (D» E Ñ ) T 2 4D = 0 (B» E, A» D Ñ ) Æ (T, D) Á» (A)-(E) ÉÆ «Í T D ÜÇ Í½É± Ø ± (A)-(E) µđº Á ÄÕÐ T = 0 (2.1) ±Ð Ò ax + by = 0 ÆÜ ½ Ð

24 v 21 «(0, 0) Í (2.1) ÐÜ D = 0, T > 0 (2.1) ÆÁ ±º º «Õ Ð «(2.1) À± Ü Å T 2 4D = 0 (2.1) ÆÕ» Ñ ½» Ñ Ð ÆÁ (A)-(E) À u 2.4 (2.1) Â ( ) λ 1 > λ 2 > 0 (source unstable node), ( ß) λ 1 < λ 2 < 0 (sink stable node), ( µ) λ 1 < 0 < λ 2 (saddle); ( ) λ 1, λ 2 = α ± iβ, α < 0 Ñ (spiral sink stable spiral), ( µ) λ 1, λ 2 = α ± iβ, α > 0 Ñ (spiral source unstable spiral)» Á Þ µ µ ÐÞ Ä (0, 0) Ö (stable)ü Ð¹ Í Ö (unstable)ü Ø Ð Ü µ Æ ÂÍ ½ µ Æ ÂÍ ÅÇ Ä ĐÅµ» µ Ñ Æ Í Æ T < 0, D > 0, (2.4) T > 0, D > 0 D < 0. (2.5) 2.2 Ò À Đ ÜÇ Æ Ç x = f(x, y) y = g(x, y). (2.6) ÄÐ ¼ (vector field) F(x, y) = ( f(x, y), g(x, y) ) Í R 2 R 2 Æ Ý (x 0, y 0 ) R 2, (2.6) ± Ð ( x(0), y(0) ) = (x 0, y 0 ) Æ Ð Ü (x 0, y 0 ) R 2 f(x 0, y 0 ) = 0 g(x 0, y 0 ) = 0. (2.6) (x 0, y 0 ) Æ dy dt = J Y, (2.7)

25 22 J Í Jacobian Ö f J = x (x 0, y 0 ) g x (x 0, y 0 ) f y (x 0, y 0 ) g. y (x 0, y 0 ) ß (x 0, y 0 ) ÆÐ ¼¾ U Å (x, y) U, (x, y) ÔÆ (2.7) Æ ( x(t), y(t) ) lim t ( x(t), y(t) ) = (x0, y 0 ), Ã Ü (x 0, y 0 ) Íµ Æ Ã (x 0, y 0 ) Í µ Æ Ç Ä À Æµ «Ð Æ³ Þ Æ «Æ Ö (asymptotically stable). Ð Æ «Ä± µ 2.1. F = (f, g) : R 2 R 2 Ð±Ã ³ (x 0, y 0 ) ÐÍ ÃÉ T, D Ð Jacobian Ð J(x 0, y 0 ) Å Þ (a) T < 0, D > 0, Ú (x 0, y 0 ) ÐÍ ÃÉ (b) T < 0, D > 0 D < 0, Ú (x 0, y 0 ) ÐÍ ÃÉ (c) T, D Æ Û»² J 2.1. (2.8) ÃÉ µù x = 2x(1 x 2 ) xy y = 3y(1 y. (2.8) 3 ) 2xy 2x(1 x 2 ) xy = 0 3y(1 y. 3 ) 2xy = 0 ÃÉ (0, 0), (2, 0), (0, 3), (1, 1). 2.1 Jacobian Ð Æ (0, 0) Ð (1, 1) Ð ÅË (2, 0), (0, 3) Ð Ü º ÐµÜ Å Á µ Ü Æ Æ Ä «(2.7) Ænullcline (isocline) N x = {(x, y) R 2 : f(x, y) = 0} (x-nullcline), N y = {(x, y) R 2 : g(x, y) = 0} (y-nullcline). Ð N x N y Í R 2 Æ ½ f(x, y) g(x, y) Æ Å ³ ĐÐ N x N y ½ (2.7) ÆÜ É³ (f, g) R ÉÆ É É³Ï (f, g) R 2 º ÝÆÉ³ É Nullcline Æ «N x R 2 ±Å É³ (f, g) ÖÒ ÉÆ ( ) N y R 2 ±Å É³ (f, g) ÐÜ ÉÆ ( ) ½ N x N y Æ R 2 Î Ë Å f g Á ÅÆ ÞÁĐÇ Æ N x N y ß Ü Î Ð ÅÆ N x Æ Å Ð ÁÆÆ ÉÉ³Ï É Å Å N y Å Å ½ R 2 ± N x N y Î Æ Â ß Ð ÁÆ É (f, g) É տ Å ր Å ւ Å ց Å 2.1 ÞÆ ÆÉ³ ÉÀ Ä x 0, y 0). É³ÕÉ ÐÍÞÆÁ Ð «Ü Á Æ (2.8) Æ É³ÕÉ Þ Nullclines x = 0, y = 0, x + y = 2 2x + y = 3 ÄÆ ÐË A,B,C,D Õ ( ) ( ) Å Ô (x 0, y 0 ) C, ± lim x(t), y(t) = (0, 3); ½ (x0, y 0 ) B, lim x(t), y(t) = (2, 0); t t (x 0, y 0 ) A Ã±¹Ð

26 23 (0,3) C D (0,3) Γ4 Γ 2 A (1,1) B (0,0) (2,0) Γ 1 Γ 3 (0,0) (2,0) 2.5 ( ) (2.8) Nullcline Þ»Ê ÊÖÊ. (µ) (2.8) Þ Ê (A1) (x(t), y(t)) µ B; (A2) (x(t), y(t)) µ C; (A3) lim t (x(t), y(t)) = (1, 1). ½Ü A Æ Õ lim (x(t), y(t)) = (0, 0). 2.1 ÄÐ (A3) ÆÕ ±Ð t ½Ü (1, 1) Ð ½ lim t (x(t), y(t)) = (1, 1)ÅÆÕ ß ±Á Æ± Ð Ò (A3) Æ Õ (1, 1) Æµ Õ (Stable orbit), ½Ü (1, 1) Ð Æ (1, 1) Æ µ Õ (Unstable orbit). Äß µ (Stable manifold) µ (Unstable manifold). ÖÏ Ò Å É³ Ï Õ Æ Þ Hartman-Grobman (Invariant Manifold) Ë Í Æ±ĐĐÞ Æ Ä Ü ¾ D Ð ÆÕ Æ ½ µ / µ Æ Ä Å¾ÐÁ ÆÉ À 2.5 Å É Þ Ì Ü ÆÕ ĐÞ Ð lim (x(t), y(t)) = (x 1, y 1 ), t lim (x(t), y(t)) = (x 2, y 2 ) t ÆÕ ÌÜ (x 1, y 1 ) (x 2, y 2 ) Æ ÛÕ (heteroclinic orbit) (x 1, y 1 ) (x 2, y 2 ), «ÛÕ (homoclinic orbit) (x 1, y 1 ) = (x 2, y 2 ). «Û/ ÛÕ ÌÕ (connecting orbits). (2.8) ÞÒ (2, 0) (0, 3) Ñ ÐÁ Ü ±Ð ÛÕ Á Ì ÛÕ Õ (separatrix, Ö separatrice), (x 0, y 0 ) Γ 1 ((0, 0) (1, 1))» Γ 2 ( (1, 1)) ½ (1, 1) Æµ Å Ð ± lim t (x(t), y(t)) = (0, 3), ½ Γ 1 Γ2 lim t (x(t), y(t)) = (2, 0). µ Ü (x 1, y 1 ) Ä «B ((x 1, y 1 )) = {(x 0, y 0 ) R 2 + : lim t (x(t), y(t)) = (x 1, y 1 )}. (x 1, y 1 ) Æ Õ (basin of attraction), Ã Γ 1 Γ2 ÐÍ (2, 0) (0, 3) Æ Í Ä Ñ Õ (2.8) ÍÐ Ù± µ (bistability) Æ Ð (1.28). Γ 3 ((1, 1) (2, 0))» Γ 4 ((4, 1) (0, 3)) ß Õ (x 0, y 0 ) Γ 3 Γ4 lim (x(t), y(t)) =, t (x 0, y 0 ) Γ 3 Γ4 lim (x(t), y(t)) = (0, 0). Ð Æµ / µ Õ Ý t

27 24 Ä 2.1 ÞÆ (2.8) µã Æ ½ ½ Ð (2.7) Ñ Á ½ (phase plane analysis). Ð ² (1) Nullcline ºÉ³Ï É ½; (2) Ü Jacobian Ö µ ½; (3) Õ «Û/ ÛÕ ½; (4) Õ ½. (2.7) Æ Õ ÍÕß T > 0 Å (x(t), y(t)) x(t + T) = x(t), y(t + T) = y(t), t R. (2.8) Æ ½Þ Ä µ (1)-(3) Í Ä É³Ï É ½ Ò R 2 + Æ Õ Æß 2.5 ÏÞ ÄÆ (2.7) Õ Æ ½ 2.3 Ü» ÓÕÑ Æ ÀÆÎ Æ ÂÖ Ó Lotka(Lotka, 1920) Đ Ó Volterra(Volterra,1925) ÆÎ x = ax bxy, (2.9) y = cy + dxy, a, b, c, d > 0. Í Æ Î (predator-prey model), x(t) Íº (prey) Æ ³ y(t) Í (predator) Æ ³ Ð Ð Þ Î ß Í Î (consumerresource model), x(t) Í y(t) ÍÍ Ð Æ Î Í x = xf(x) h(x, y) (2.10) y = yg(y) + r(x, y), f, g Í x, y Æ Ç h(x, y), r(x, y) R 2 + Æ ÄÝ º Ú Æ± f g 1.4 ÏÞÆ Ç ÎÇ ÖĐĐ º ½ ß º g(y) = d Ø g(y) = d 1 d 2 y, ½ h r Æ Ç h(x, y) = kr(x, y), (2.11) ½º Ú±» ± ½ (2.11) Þ h(x, y) À Ç h(x, y) = φ(x)y, (2.12) φ(x) ÔÍ 1.4 ÏÞ Æ Ý (predator functional response) ½ º ³±Ð Ü Æº 1.4 ÏÞÆ Ý Ï (2.11),(2.12) Æ Î x = xf(x) φ(x)y y = yg(y) + kφ(x)y. (2.13) ³ Þ (2.13) º Rosenzwig-MacArthur Î (Rosenzwing-MacArthur, 1963), Ë 70 Rosenzwig, May ¼Ó Ó (Science) Æ ³ µ (2.13) Æ ¼ «

28 º Ó Æ ±Å (2.13) Æ Ó ÖÏ ½ ÄÆ 2.5 Ï (2.13) Ã ½ ±Þ (2.13) x = ax(1 x N φ(x)y y = dy + kφ(x)y, 25 (2.14) Þ a, d, N, k > 0, ½ φ(x) Í 1.4 ÏÞÑ«Holling II Ý Å ÄÕÐ (2.13) (2.14) Æ Þ y ½ ÅÐ Î (1.29) Ð Í ³ Î Æ Ø ³Æ ± y(t) Í Ð «(1.29) Í (2.13) ÆÐ Ð¼ Òµ (2.13) Æ Î (Beddington, 1975) (DeAngelis 1975) µ Beddington- DeAngelis Î (2.15) Þ a = 0, Ã x = rx(1 x k ) mxy a + by + cx y ǫmxy (2.15) = µy + a + by + cx. x = rx(1 x k ) mx b + cx/y y = µy + ǫmy (2.16) by/x + c. ÍÐÐĐ - º ³½ÔÆ Î (ratio-dependent model)(arditi-ginzburg, 1989). Î (2.10) Þ ± ¼Ð Ð ÚÐ ± Ð Æ (interaction) Þ Ú ÝÆÎ Ñ ÐÎ (competition model), (2.10), Ä Ð ÐÎ x = rf(x) h(x, y) y = yg(y) r(x, y). (2.17) ĐÐ (2.8) ÔÍ (2.17) ÆÐ ± «Ð Ñ ÆÐÍÐ Ð (interspecies competition), «Ð Â «ÆÐ Ð Ð (intraspecies competition). Ð Logistic Đ±Ð ÐÆÐ Ç ½ ³Ç Ð Ç (2.17) Þ f, g Ñ«Logistic (2.17) µð Ð ² µð Ð ÐÎ Ó ± ¾ Ð Å ßÌÓ ± ÈÌß ÅßÑ ± Ð Æ Þ ½ x = rf(x) + h(x, y) y = yg(y) + r(x, y). (2.18) Ð Î Æ Î ±Î (cooperative model, mutualism model). Đ Ó Volterra 20 Ë ¹ Æ Ç±µ Ò Ð ÒÆ Ð x = x(a + bx + cy) (2.19) y = y(d + ex + fy),

29 26 a, b, c, d, e, f R. ³ ĐÐ±Å Ô Æ (2.19) ¹ (2.19) Lotka-Volterra ¾Ð ÆÐ ÇÍ x = rf(x, y) y = yg(x, y). Î Ä f g (x, y) < 0 (x, y) > 0. Ð Ç Kolmogorov Î y x (2.20) Ç Ä Å Ó Î Lotka 1920 Æ ³ÞÔ (2.9) É µ ÇÐ Æ Ó A + X 2X, X + Y 2Y, Y P, (2.21) A º ÆÜ X º Y P Ô µæ A Æ ³ Ô Ô Å (2.9) Ð» ÔÍ Ó ma + nb C (2.22) Æ [A] m [B] n ½ [A], [B] Í Ó¼Ü A, B Æ ³ ± m, n Í ½ m A n B Å Ð m + n Í Ð ÆÍ (order). ¼ ÓÞ¾Ð À ËÍ Þ (autocatalytic reaction) Þ Í Ó Å ³ Æ ¼ ½ Þ ÍÅ µ Ð Ó¼Î» Æ Ó¼Ü Ð ± ³ Æ Ó¼Ü Ð Þ Ó ma + nb k1 (n + p)b, (2.23) m, n, p N. Ð É A = mk 1 A m B n, B = pk 1 A m B n. (2.24) (2.23) ÞÆ Í Æ ½ ma + nb k1 k2 (n + p)b. (2.25) Ã ÆÎ A = mk 1 A m B n + mk 2 B n+p, B = pk 1 A m B n pk 2 B n+p. (2.26)» Ð Ó ÆÎ ºÅ (Schnakenberg, 1979) Þ Ó 2X + Y 3X, A Y, X B, (2.27) A B ³ ³ Ã x = [X], y = [Y ], a = [A], b = [B]. ¾± Î Í x = x 2 y x + b, y = x 2 y + a, (2.28)

30 27 (a) Ó Prigogine Æ Brussel ÓÐ (Prigogine, 1978) Ó A X, 2X + Y 3X, B + X Y + D, X E (2.29) ÐÆ Brusselator Î x = a + x 2 y (b + 1)x, y = bx x 2 y, (2.30) (b) Ó Gray Scott(Gray-Scott, 1983, 1984) Æ Ó U + 2V 3V, V k P (2.31) «³ F Æ U º µ ½«³Æ U V º ÒÐ ÝÔ Å U = UV 2 + F(1 U), V = UV 2 (F + k)v. (2.32) Ç Ó Î Þ (reaction rate) ÐÒº»ÁÐÅ Æ ¼ ³ ½ Ø Ä ÅÆ Þ ÍÛ±ºÆÇ Æ Â Þ ¼ ³ Ð (Lengyel-Epstein, 1991) µ CIMA Ó A k1 X, X k2 Y, 4X + Y k3 P, (2.33) k 1, k 2 ½ k 3 = k 4 /(k 5 + X 2 ) Ø Ä± x = k 1 k 2 x 4k 4xy k 5 + x 2, y = k 2 x k (2.34) 4xy k 5 + x 2. º ¼Ø² ÐÐ (activator) ÐÐ Û (inhibitor). (Gierer-Meinhardt, 1972) ÐÐ Ð - Û (activator-inhibitor system) a = ρa 2 /h µ a a + ρ a, h = ρa 2 µ h h + ρ h, (2.35) a(t) Í h(t) Í Û ρ a, ρ h Í±Ð Ó¼ÜÎ ½ µ a, µ h Í±Ð Ó¼Ü a 2 Æ ±Ð Ó¼ÜÆÎ ½ Ð Þ h 1 Æ µ Û± Ä ± «ÆÎ»Ï Ö Ó Hodgkin Huxley (Hodgkin-Huxley, 1952) Ò ¹Å Õ ÆØÞ ÐÐ Õ ³ Ý Î Õ ÞÆ ± (action pontential). (FitzHugh, 1961) (Nagumo, et. al, 1962) µ Hodgkin-Huxley Î Ðµ FitzHugh-Nagumo Î V = V V 3 /3 W + I, (2.36) W = a(v bw + c), Þ V (t) Í ± Æ W(t) ÍÐ Ã ³ I Í Ù ÆĐÎ a, b, c Í Å

31 28 Å Ä Å ĐÓ (Klausmeier, 1999) µ Ë Ó¼ ÆÎ w = a w w n 2, n = wn 2 mn. (2.37) w (t) ÍÐÆ ³ n(t) ÍÓ¼ ³ a ÍÇÐ³ m ÍÓ¼Ô, w ÆÍÐ ³ wn 2 ÆÍ ÐºÓ¼ Æ³ wn 2 ÍÓ¼ ÇÕÐ ÄÆ Ð ºÎ ß µ Ä Ð Ù ½ ĐÎ ÞÆ¼ Ü Æ»± 2.4 Ñ ÊÒ¾Ø (nondimensionalization) ÓÎ ÞÆ ³ (variable) Å (parameter) Ù±Ð ³Ì (dimension) ½ÐÂ ³ ³Æ ³»»¼ ³Ñ Æ Ö ÛÞ±Đ»¼ ³ ± (L), Ü³ (M), (T), (I), Ó²± (Q), ¼ÜÆ³ (N) ÐÅ± (J). ¼Ð Î Þ ¼ Æ ß Đ Ð»¼ ³ (S). Ð Î ÞÆ / ³Æ³Ì Í»¼ ³Æ Ð³ 2.2. Lotka-Volterra «Đ x = x(λ ax by), y = y(µ cx dy). Ú Å (2.38) t T λ T 1 x S 1 a T 1 S1 1 y S 2 b T 1 S2 1 µ T 1 c d T 1 S1 1 T 1 S2 1 º Æ ÍÍ ½ ß º µ ß x Å y x Å y Å³ µù ÓÚµ Õ q µ dim(q)å (a) z Å t µ Z Å T, Ú dz/dt µ ZT 1. (b) A = B, Ú dim(a) = dim(b). (c) A ± B Ý ÙµÙ Ú dim(a) = dim(b). ³Æ³Ì º Æ Å Æ Ä ÅØ «³ Þ ³Æ³Ì 1, ½Ü ³Ì (dimensionless). Ä 2.2 ÞÆ ÚÉ ÐØ - ³Ì (nondimensionalization).

32 29 Å «s = λt, X = ax/λ, Y = dy/λ, Ã³ Û (s, X, Y ) Þ Ü ³Ì ³ Ç Ãdx dt = dx dx ds dx ds dt = λ dx a ds λ = λ a X(λ λx bλ d Y ) dx ds = X(1 X b d Y ). dy ds = Y (µ λ c a X Y ). A = b/d, B = µ/λ, C = c/a, Ã (2.38) À Ä²Å x, y ½ X, YÅ x = x(1 x Ay), y = y(b Cx y). (2.39) ĐÅÅØ±µ ³Ì Æ (2.38) ÞÆ 6 Å Åº³Ì (2.39) Æ 3 Å ½ Å Í Å Æ (2.39) ±Ë Å ³ ± ÐÁ ½ ÞÆ ± ÞÆÎ Æ Ó Í³Ì ½ÞÆ Buckingham π Ä ÖÏÆ ĐÙ Ð Ç ÆÛ«: Ð ÓÎ À Þ²Û n ¼ ³ Í ³ß Å Å ½ n ³ k Æ»¼ ³ É ÅØ Ð ²Û n k º³Ì ³Æ Buckingham π ³ µ ÝÆº³Ì Æ Æß Í ÝÆº³Ì Í ÐÆ Ò ÅØ Æ Þ ÒÂ Æ Å ÞÆ Ç 2.3. Rosenzwig-MacArthur Ú x = Ax(1 x N ) Bxy C + x, y = Dy + Exy (2.40) C + x. µù ½ t T A T 1 x S 1 N S 1 y S 2 B S 1 S2 1 C S 1 D E T 1 T 1 Í µ Ú (2.40) µ s = Dt, X = x N, Y = By NA, dx ds dy ds axy = ax(1 X) b + X, dxy (2.41) = Y + b + X,

33 30 ß a = A/D, b = C/N, d = E/D. º Ì ¼Í Ú (2.40) µ s = At, X = x C, Y = By CE, dx ds = X(1 X k ) mxy 1 + X, dy ds = θy + mxy 1 + X, (2.42) Æ k = N/C, m = E/A, θ = D/A.(2.41) Å (2.42) ÐºÇ Buckingham π» µù Å³Þ ÙÞ Ñ Ð (2.40) Ú N Å ÚÞ ± D Ù (2.41) (2.40) A Å B Ù (2.41) µí«a. (2.42) Ý (2.40) ÚÞ ÛÕ A Å Â C ÙÌ³ Î B Å E Íµ m. ÓÎ Æ ³Ì Ð Í Ð Ã ± ³ ½Æ ÐÁ 2.5 À Đ ÊÕÆÆ ÂÛÇÕÞ» (2.6) Þ ÜÜ Æß µ Ð Æ Ã Ä ÒÌ ÐÜ ÅÜ Æ Ç Ñµ Æ 2.4. º Lotka-Volterra «Đ (2.38). x nullcline : x = 0, 1 x Ay = 0 Å y nullcline : y = 0, B Cx y = 0 Æ ÃÉ (0, 0), (1, 0), (0, B), ( 1 AB 1 AC, B C 1 AC ). µ x Å y Ð³ º x 0, y 0 Í ÃÉ x 0, y 0 Ï A, B, C. ß º A Å C, B µ È ³ Í (a) 1 AC < 0, C > A 1. (2.38) Æ A Å C Ð³ µù «Đ xy AC > 1 «ĐÊ º Ö µê«đ (Strong Competition). x x A 1 C B C A 1 B 2.6 Ñ Ï (2.38) B Æ ( ) AC > 1 «Ñ, (µ) AC < 1 Ñ º 2.6- ( 1 AB 1 AC, B C 1 AC ) È µ 1 AC < 0 C > A 1, Ú Ù A 1 < B < C Ù «B = A 1 Ù Ð (0, B) Ç «B = C Ù Ð (1,0) Ç ßÊº Æ 2.6- È À Jacobian Ð Æ Æ (0, B) Ù B < A 1 Ùµ Ù B > A 1 Ùµ ; Å (1, 0) Ù B < C Ù Ù B > C Ù Ù A 1 < B < C Ð ÛÐÑß Ð 2.1

34 31 y y x x y y x x 2.7 Ñ Ï (2.38) Â ( ) B > max{c, A 1 }; (µ ) 0 < B < max{c, A 1 }; ( ) C < B < A 1 ; (µ ) A 1 < B < C. (b) 1 AC > 0, C < A 1 ßÐÉ (Weak competition) Í B = C Å B = A 1 µ Ý È ÅÐ Ù µ A 1 > B > C, Ë Ð Ù A 1 > B > C Ù ³ ÃÉ (1, 0) Å (0, B) Ð Ù È Ú 2.6- Ý Ä» ½µ Lotka-Volterra ÐÎ (2.38) Æ Òµ AC = 1 Æ «Ä Å Ä ÅÅ B = µ, ½¼Ð y»¼ð x Æ Ç Æ½Ô λ ÄÆ ½ÒË B < min{a 1, C}, ½ y Æ Ç» x Á½ËÎ Ã y ÊÝ ½ x ß ± Å (1, 0)Å B > max{a 1, C} Ã Ñ y ß x ÊÝ B Ô Þ Ô ±Ð ßÜ Í Ð»ÅÐÆ «± «Ð ß µ ½ÅÐ ß Íµ Æ ß Í Ðµ ÆÜ É Æ Ü ½ÆÉ±ÒËµ ĐÓÞÆ À (principle of competition exclusion). Ð Ãº ß (x, y ) x 1 + y 1, Ã ßÍ Æ Ð ± B Ð ½¾Ð ÊÝ ĐÅ x + y B = 1 Í Ì± Ü ÆÒ ßÔÍÒ ÐÅ ±Ð ³ Ð Æ¾Ð ÜÐÞ Ð Ð ÃÍ»Ö ¼Ó Gause 1934 ÐÆ Ø ¼ ÓÞ²± ß± ÁÛ Û Ð Ã ÄÕÐ Ç 2.4 ÞÆÜ (1, 0) (0, B), Üµ µ Æ Ò Í Ü Æ Jacobian Ö J Æ Ç D = 0 Æ Ø J Ù±Ð ½ Ô Ð Æ ÂßÏ Ð «Í Ä ÅÇÍÏ Ð Â ± Í

35 32 ¾Ð Æ Jacobian Ö Æ T = 0 Æ T = 0 ½ D > 0 Ã J Ù±Ð Ô Ø Ä± 2.2. (Hopf ÓÄ ) Á λ R Ã x = f(λ, x, y), y = g(λ, x, y), (2.43) ÕÕ f, g µ±ã ³Â (2.43) Í ÃÉ { ( λ, x(λ), y(λ) ) : λ λ 0 < ε} Ü x(λ), y(λ) ±Ã ³ x(0) = x 0, y(0) = y 0, Ë Jacobian Ð J(λ) = D (x,y) (f, g) µ µ(λ) ± iω(λ), (i) µ(λ 0 ) = 0, ω(λ 0 ) > 0; (ii) µ (λ 0 ) 0. (2.43) Ù (λ 0, x 0, y 0 ) ¾ Í Å { ( λ(s), x(s, t), y(s, t) ) : lim λ(s) = λ 0, lim T(s) = s 0 + s 0 + 2π ω(λ 0 ), lim s 0 +(max ( x(s, t) x 0 + y(s, t) y 0 )) = 0. t R 0 < s < δ} Æ Åµ T(s), Ë 2.8 ( ) λ < λ 0, (ß): λ = λ 0, (µ) λ > λ º Klausmeier (2.37) ÃÉ Å Å È ÃÉ (a, 0), (w +, n + ), (w, n ), Æ (w ±, n ± ) µ 2m w ± = M ± M 2 4, n ± = M ± M2 4, 2 Ùß M = a/m. (w ±, n ± ) Ù M 2 Ù Ù Ù M = 2 ÙÍ Ë Ðµ ( ) 1 n 2 2wn J =, n 2 2wn m (a, 0) Ð Ù (w, n ), M > 2, Ú Æ µ 1, m. Ù (w ±, n ± ) º ( ) Mn± 2w J(w ±, n ± ) =, Mn ± 1 m Det(J) = m ( 2M M + M 2 4 2) < 0, È± Jacobian

36 33 (w, n ) Ù M > 2 Ù µë ¼Íµ Ù (w +, n + ) M > 2, Ù (w +, n + ) Det(J) = m ( M M + M ) > m ( M 2 Tr(J) = Mn + + m = M(M + M 2 4) 2 2 2) > 0, ½ h(m) = M M + M 2 4, Ú É h Ù M > 2 Ù±Ã ³ M > 2, h (M) > 0, «2 m < 2 Ù Tr(J) < 0 (w +, n + ) ÝÆ m > 2 Ù Ù Í M n > 2 Ü Tr(J) = 0, (w +, n + ) Ù M > M n ÅÙ 2 < M < M n Ù Ù M = M n Ù Det(J) > 0, Tr(J) = 0, µù Í ĐÁ ÛÑÙ M = M n ¾ Í» α(m) ± iβ(m) Ü α(m) = Tr(J) ( ) 2, [α(m)] 2 + [β(m)] 2 = Det(J), α (M n ) = 1 2 Tr(J) = 1 4 h (M n ) < 0, Hopf È Û Ù M n Í Å ( M, w + (M), n + (M) ) ÈÝ + m. n (w +,n + ) (w,n ) 2 w 2.9 Klausmeier Ï (2.37) ( ) 0 < m < 2; (µ) m > 2. Hopf ÅÎ À Õ Å Õ ß Æ¾Ð Poincare-Bendixson Ä Ð Ð ± ÆÂ Þ 2.3. Ã (2.6). ÕÕ (f, g) ±Ã ³ (1) Σ 0 = { ( x(t), y(t) ) R 2 : t R} Ð (2.6) Í Å Ú Σ 0 ÏÁÍ ÃÉ (2) O R 2 ÐÍ O ÏÁ ÆÃÉ Ù O O Ñ Ó (f, g) O Ú O ÁÍ Å (3) Σ 1 = { ( x(t), y(t) ) R 2 : t 0} Ð (2.6) Í Ú«t Ù Σ 1 Đ²Æ Æ Í (a) Í ÃÉ (b) Í Å (c) Í / ±Ð Ç (4) (Dulac Ú) O R 2 ÐÍ ± B : O R ÐÍ ±Ã ³Â Ü (Bf) x + (Bg) y ÄË Êµ» Ú Æ Å Ù O Poincare-Bendixson» ÇÆ LaSalle Ô Å 2.4. Ã (2.6), O R 2 ÐÍ ±

37 34 (a) Ù V : O R ÐÍ ±Ã ³Â Ü V (x, y) C, C R, Ë Æ (2.6) ( x(t), y(t) ), d dt V ( x(t), y(t) ) 0; (b) O (2.6) Ð (x 0, y 0 ) O, Ú (x 0, y 0 ) Ý± ÝÆÙ O. (x 0, y 0 ) O, (x 0, y 0 ) Ý± Đ²Æ ß V (x, y) = V x dx dt + V y dy dt = V x {(x, y) O : V (x, y) = 0}, f(x, y) + V y Ä ±Æ Å 2.3 ÞÆ Î g(x, y) Rosenzwing-MacArthur Ú x = x(1 x K ) mxy 1 + x, Ä x 0, y 0. Ü ± (0, 0), (K, 0), (λ, V λ ), Þ λ = y = θy + mxy 1 + x, (2.44) θ m θ, V λ = (K λ)(1 + λ), Km ß (λ, V λ ) ÐË Æ ÂÍ 0 λ K. λ K (K, 0) Í Õµ Æ ½ (x 0, y 0 ) R 2 +, lim t ( x(t), y(t) ) = (K, 0). 0 < λ < K, ³ ÜÐ 0 < λ < K, Det(J) = Í µ Æ K 1 2 λ(k 1 2λ) J(λ, V λ ) = K(1 + λ) K λ K(1 + λ) θ 0 θ(k λ) λ(k 1 2λ) > 0, ½ Tr(J) =, (λ, V λ ) 0 < λ < k 1 K(1 + λ) K(1 + λ) 2 < λ < K Íµ Æ Ä Dulac Ã Ë K 1 < λ < K ÐË À± Õ (Hsu-Hubble-Waltman, 1978): 2 x h(x, y) = ( 1 + x )α y δ, Þ α, δ R Ø Dulac Ã ÞÒ Ä Ï Æ α, δ R, Ë (fh) x + (gh) y 0, (x, y) R 2 +. Ü½ 2.3 ÞÆ Dulac Ã Å (2.44) K 1 < λ < K À± Õ Ø (λ, V λ ) Íµ 2 Æ (2.44) ß ÛÕ Ø 2.3(3) (x 0, y 0 ) R 2 +, x 0 0, y 0, ± ( ) x(t), y(t) = (λ, Vλ ), ½ (λ, V λ ) Í ÕÃ µ Æ Õ Í (0, 0), (0, 0) Í µ lim t (K, 0) Í Ðµ Õ x- 0 < λ < K 1 2 (λ, V λ ) Í µ Æ Ø 2.3, (2.44) ÐË ±Ð Õ Äß Hopf Ð Õ ß Õ Æ Ð µ ÍÐ Æ

38 (2.44) (2.44), ( ÖÑ, 1981) Ëµ 0 < λ < K 1 2, Ð ÕÍÒ (λ, V λ ) Õµ Æ Å (Î³ 1986), (Kuang-Freedman, 1988) Æ (2.44) Lienard Lienard Õ Ð Ëµ 2.44 Õ Ð ÐÄÆ Ë Ð Æ Î (2.13) (2.44) Â ( ) (k 1)/2 < λ < k; (µ) 0 < λ < (k 1)/ Æ Ðµ Õ Ð (limit cycle), ÒË Rosenzwing-MacArthur Î Î µ Æ ß λ = θ, K( ) ± Å Ý K < λ m θ Ê ½º ß ½ λ < K < 1+2λ È½ º µ ß (λ, V λ ), Í K > 1 + 2λ Ðµ º µ ½ ßÜ ³ º Ð Ù¼ÐÊ Ø (Rosenzwing, 1971) Ð Ë Ð ÈÈ (paradox of enrichment). 2.6»³Ï Á Î (2.6), Ä»Î x n+1 = f(x n, y n ) y n+1 = g(x n, y n ). (2.45) Ð Á Æ µ ºÅ Í ÆÁ ½ º Å (2.43), Ø (2.45) Æ Õ ½ Æ Ù ÌÁ «Ð Î Ë (2.45) ÞÐ Ø Ä

39 (Hardy-Weinberg &v) {} F Mo Hardy(Hardy,1908) M FVs Weinberg (Weinberg,1908) y b` T X} b[&v b` (genetics) Za7us1 U a7d (phenotype) m #{U6G}hb`D (genotype) g a7d b`d}g :9b`D}g + b` (Allele) O#s "hu b`h' +b` A M a, #b`dh AA, Aa, aa k AA M aa r) b`d (homozygote), Aa r)_ b`d (heterozygote). UU b`dpmy ` Z a7d uu9s1 _ A9b`D \":hg hg{us1 _ Z su U X} B8.; ^ n s1 {Ub`D +b` A \") p, a \") q, v p + q = 1. hgqq Gb\" %4 `4}gW lb`d zj} B8 (p, q ) 6 n n n (2.46) n n n 1 pn+1 = p2n + 2pn qn = pn 2. 2 qn+1 = q + 1 2pn qn = qn n 2 (2.46) i} Hardy-Weinberg &v :9 +b`ub`d \" X}Xwg_ (2.46) ` Æ 9U Æ B E,LP 9 z }Y 9 1 [ \$ { ;I \G 9 AA, Aa, aa - \$ { ;B = S, S, S, 1- \$ Æ l Bw; p, 2p q, q. E\ } 9 = (2.46) n 3 n n 2 n (S1 p2n + S2 pn qn )pn S1 p2n + 2S2 pn qn + S3 qn2. = 1 pn+1 pn+1 = qn+1 (2.47) < U \$ q o z Æ \$ / < * \$ \$ A U U ", a(advantageous), \G S = 1, S = S = 1 + S, (2.47) F (2.47) ^ Fisher-Haldane-Wright pn+1 = pn + * A U (recessive) Spn (1 pn )2, 1 Sp2n ", a(advantageous), \G S1 = S2 = 1, S3 = 1 + S, (2.47) pn+1 = pn + X (dominant) (2.48) ` Sp2n (1 pn ), 1 Sp2n (2.49) \ S U}Y VÆ f g \$ 2.4 l {pn} U O j+ n lim pn = 1, \ ns " \$ \$ A! a \ Hardy-Weinberg 6 K U Æ 2.12 Fisher-Haldane-Wright F =drr % u d (#) (2.48), ()) (2.49)

40 37 ÄÕÐ (2.47) º É dp dt = p(1 p) (S 1 S 2 )p + (S 2 S 3 )(1 p) S 1 p 2 + 2S 2 p(1 p) + S 3 (1 p) 2. (2.50) (2.50) Þ S 1, S 2, S 3 Ã S 1 p 2 + 2S 2 p(1 p) + S 3 (1 p) 2 1, (2.50) dp dt = p(1 p)[(s 1 S 2 )p + (S 2 S 3 )(1 p)]. (2.51) (2.50) (2.51) ÐÍÆ ½ S 1 S 2 = S 2 S 3 = S, Ã (2.51) Logistic ÌÁ ÍÅÇÞ Æ Æ» Fisher Æ Å Ä ĐÓÞ ÀÆ Í Nicholson-Bailey ¼ - Û Î (parasitoid-host model): N n+1 = N n exp (r ap n ), p n+1 = N n ( 1 exp( apn ) ). (2.52) Þ µ ¼Æ Å Beddington Î N n+1 = N n exp ( r(1 N n K )), ( p n+1 = N n 1 exp( apn ) ). (2.52) (2.53) 20 Ë 90 (Constantino, Desharhais, Dennis, Cushing, 1997) ÐÐ É Ç Ú (flour beetle) ÆÎ L n+1 = βa n exp ( c 1 A n c 2 L n ), P n+1 = L n (1 δ 1 ), (2.54) A n+1 = P n exp ( c 3 A n ) + A n (1 δ 2 ), L n Í P n Í ±» Ñ Æ A n Í (2.54) Æ Ô»ÁÛ Ø ËÐ Ü½ Ë µð ĐÓ ß Ð 2.7 Î»Ð 1. dy dt = a R, Ü º ( ) ( ) a 1 x(t) Y, Y =, a R, 1 0 y(t) 2. x = x(2 x y). y = y(y x 2 ) Ü x 0, y 0 ¾Á Ð nullclines, Ü Jacobian Ö separatrix, pplane ÐÁ

41 38 3. Æ Î x = ax(1 bx + cy) y = dy(1 ey + fx). ÐÐ º³Ì 4. Æ Î x = x(1 x) + Axy y = y(b y) + Cxy Æ Ë ÐÜ Æ 5. Schnakenberg Ó x = x 2 y x + b. y = x 2 y + a (a) ÐÜ Ò µ (b) Matlab Ð (a, b) Å ÜÇ Ü µ µ Æ (c) a = 2b, Ð Hopf MatCont Ü É º Õ (d) Ë Æ ± Ü½ Ë Ü µ Í ± Õ ( É (x+y) = a+b x) Æ : Ë m < 2 À± (a) Ë Æ ± (b) pplane, MatCont º

42 ÍË ± ÎÐ 3.1 ±ÏÑ Æ» (diffusion) Í¼Ü ( ) Õ Æ Ç Ð Ò Æ»Å¼ÜÜ ±ÍÆ»Å ± Æ Æ»ÍÐ ¼ Ø½ ÓØ ½¼ÜÆÜ Õ ØÞ ÐØß Í ³ ± ¼ÓÞ Æ»ºÉ ÐÐ ÆÕ Ç ¼Ü Ð Ç ± Àº ÒÆÆ» ËÍ Õ Ø (heat transfer) Æ» ¼ (conservation law) Fick Ã D Ð ¾ ½ D R n Í Ð u(t, x) ÐÐ¼Ü t Ú x ÔÆÅ± ( ±) Ý Å±Ý Ð Ý «x D «(O n ) Ð ²Û x Æ ¼¾ O n O n+1, m(o n ) 0(n ) m(o n ) ¼Ü O O n Æ ³ (É±) Å±Ý n ÞÆ³ u(t, x) = lim Ä Ò Ð Í ÖÏ n m(o n ) ß Đº Ð «Í Æ ¼ÜÆ³ Í ³ Ü³ Ð»¼ ³ O D ¾ O ÞÆ³Í U(t) = u(t, x)dx. Ä U(t) Þ Æ Á Ý ÐÆ Â du dt = d u u(t, x)dx = (t, x)dx. dt t ¾Ð Ç U(t) Æ ¼ÜÜ O Æ ÐÜ du dt = J(t, x) n(x)ds, O O O J(t, x) Í¼Ü x ÆÉ Å³ (flux) ÍÐ É³ ½ n(x) Í x Æ É³ J(t, x) n(x)ds µü O Æ Â µ O Æ¼ÜÆ Å³ Í U(t) ÆÇ ³»± O Ä Å ¼ J(t, x) n(x)ds = O O O O div(j(t, x))dx. u (t, x)dx + div(j(t, x))dx = 0. t O Ý u t (t, x) div(j(t, x)) ¾ O Æ³ ½ Ð ½ Ä Å» u (t, x) + div(j(t, x)) = 0. (3.1) t Fick Ã (Fick s law) ÔÍ ¼Ü Í ±Ô É ±Ô Æ Ó Ç ÄÐ Ð Ý u : R n R u ÆÔ Ç Æ ÉÍÄ ± (gradient) É Fick ÃÍ J(t, x) = d x u(t, x). (3.2) 39

43 40 x Í x ³Æ ± Æ (3.1) (3.2) Á div( u) = u Ä Å Æ» Þ u t = d u(t, x), t > 0, x D, (3.3) u = 2 u x u 1 x u 2 x 2 = n Í Laplace Ü d ÍÆ» (diffusion coefficient) Æ» ºÐ ÐÆ Ç Ð Ð Æ» ½ D = (a, b) Ð D = R n Á Æ R» Í x ÆÏ ¼Üßº R ÆÏ Þ² Å, ¼¾ u(t, x) t Ï x ÔÆ Á É É ÆÉ Þ 1 2 Ã n k=1 2 u x 2 k u(t + t, x) = 1 2 u(t, x x) + 1 u(t, x + x), (3.4) 2 t Í ÐÚÆ t x Î Ä Taylor ÇÆ (3.4) ÞÆ³É Ô Å (3.4) º ÍÍÆ½ u(t + t, x) = u(t, x) + u (t, x) t + ÍÍÆ t u(t, x ± x) = u(t, x) ± u x (t, x) x u 2 x 2 (t, x)( x)2 + ÍÍÆ. ÍÍÆº ± «Æ ÂÍ Ä ÅÐ Æ» u ( x)2 2 u (t, x) = (t, x). t 2 t x2 t 0, x 0, ( x) 2 lim t, x 0 t = d > 0. u t (t, x) = u d 2 (t, x). (3.5) x2 Ð ÇÍ Þ² Á (random walk) ÆÉ Đ Í Brown ÆÆÐÐÐ Ç Æ R ÆÏ R n ÆÏ Äß n ÆÆ» (3.3) É É Æ É p q(p + q = 1) Ã Ä Å u t (t, x) = u d 2 x 2 (t, x) u u 0 (t, x), (3.6) x u 0 = lim t, x 0 (q p) x. t (3.6) ÍÆ»Ù (diffusion-advection equation) u 0 ÞÐ ÉÙ ÆÚ± Å ÄÕÐÆ» d Æ³Ì L 2 T 1 (L ± T ) d ÆĐÎ ÁÛÉ Ð d Þ ²± ÜÆ ½

44 ½ Ü²³ ¼ÜÆÕ ¾ Å Ä R n ÞÆÕ Æ» Æ» (fundamental solution) u t = d u, t > 0, x R n, u(0, x) = δ 0 (x), x R n, (3.7) δ 0 (x) Í Dirac-Delta Ý ½Ñ«Ý x = a, δ a (x) = 0 x a, R n δ a (x)dx = 1. Ø (Fourier ) (3.7) Æ u(t, x) = 1 (4πdt) n 2 e x 2 4dt. (3.8)» ß Ô u t = d u, t > 0, x R n, u(0, x) = u 0 (x), x R n. (3.9) Æ 1 u(t, x) = (4πdt) n 2 R n e x y 2 4dt u 0 (y)dy. (3.10) Æ» Æ» ß ºÐ Ç Ð ¾Ð ± Æ ÍÙ ÐÏÆ random walk ÆÄ ÆÅ É ½ Ä Ð Ü O Ð Á É É Ð Á n Á Å Å m Æ ±ºĐ Đ n m Æ Í ³ ĐÐ Ø ÙÐ Ú À (binomial distribution) Æ»É À: P(n, m) = 2C m n (1 2 )n, P(n, m) Í n ÚÅÚ O m ÆÉ, 0 m n. É ÞÆÞ Ð, Ú À Ó Å ÞÔ 0, Í 1 Æ Ó À (Normal distribution): f(x) = 1 2π e x2 /2. ÌÁ Ä ĐÅ n = 1 Æ» ÍÞÔ 1, Í 2Dt Æ Ó ÀÝ ½ Í 2Dt Æ Ú±Ç ÒËµÆ»ÆÅ± ÀÞ.» Æ ¼Æµ (biological invasion) Æ. Ä Þ n = 2 Æ «. Ð ¼Ð µð ÞÅ±Æ ÆÔ Đ ÍÐ δ Ý, ½ ¼Ð Å± Ç, Ä± Malthus Æ» : u t = u d( 2 x u y 2 ) + au, t > 0, (x, y) R2, u(0, x, y) = Mδ (0,0) (x, y), (x, y) R 2, (3.11)

45 42 a > 0 Í Ç, M Í ¼Ð Æ ³. (3.11) Æ ½ Ð¼ÐÆ Õ Ç u(t, x, y) = M x 2 +y 2 4πdt eat 4dt, (3.12) R 2 u(t, x, y)dxdy = Me at. (3.13) Ä ± Æ Ç¼Ð ËØ Æ ¾ (0, 0), ¾ 0, ½ t > 0, Ç¼Ð Ë Ø ¾ Ð R(t) Æ. Ä Ð ¾ ÂÆ ¼ ³Æ M, Ã Ý Å R 2 B R(t) u(t, x, y)dxdy = Me at e R 2 (t) 4dt = M, R(t) = 4adt, A(t) = 4πadt 2. (3.14) A(t) ÍÇ¼Ð ÆµÆ ¾Ç³. ¾ 4ad ÆÚ±ÆĐ. Ð 1951 º (Skellam, 1951) Â º (muskrat) Æ»ÆÚ± Ç³ (km 2 ) Ê À Ë² (3.14) Ø, ad 32.15, º ß ¾ÆÇĐÚ± R (t) = 4ad = ( / ). º ÅÉ ¼ß ¾ÆĐÎ. ÇÕÐ Å±Đ, Õ ÇĐ Ï, Ð Í Ðµ Æ. R (t) = 4ad ÂÌß (traveling wave) Æ Đ. n = 1 Æ Malthus u t = du xx + au, t > 0, x R, Æ Ç u(x, t) = v(x ct), Ã v(z) dv + cv + av = 0, z R. v(z) ÍÐ, Ø c 4ad. ( c < 4ad, Đ). Ä ÎÚ c = 4ad, v(z) = c 1 e az + c 2 ze az. v(z) 0, c 2 = 0, v(z) = c 1 e az. ½ u(t, x) = c 1 e a(x 4adt). Ð ÒË Ç Æ, «ß ÐÎ Æ, Ð Íº Æ. ÅÇ Ä Logistic Ç Malthus Ç Ü Ð.

46 Ø ÂÇ Ç Ð ¼Ð (habitat). Í± Æ, Ø Ä Í R n ÞÆ Å ¾ D. Í Æ, Ä u = 0. Ã u t = d u + au, t > 0, x D, u(0, x) = u 0 (x), x D, (3.15) u(t, x) = 0, t > 0, x D, D Í D Æ. ³ u(t, x) = U(t)V (x) U (t) = ku(t), t > 0, d V (x) + av (x) = kv (x), x D, V = 0, x D, Ã λ = (a k)/d Í Laplace Ü Æ Ô, V (x) = λv (x), x D, V (x) = 0, x D, Ô Ð (3.16) ±Ð Ô 0 < λ 1 < λ 2 λ n. ÌÁ Å Ä Ð Ð Æ ( ËÀ (Ã Ü, 2009)). (3.16) Þ, 3.1. u(x) + c(x)u(x) = λu(x), x D, u(x) = 0, x D, (3.17) ß c(x) Ð D Ñ ±ÃÂ. Ú (1) (3.17) µ {λ i (c) : i 1}, λ i (c) < λ i+1 (c), lim i λ i (c) =. (2) λ 1 (c) ÐÍ, «λ = λ 1 (c), (3.17) ÐÍ, Ë Í Â ϕ c 1(x) > 0 Ý. (3) (Courant-Fischer) λ i (c) = max min R c (u) = min max R c (u). S i u S i S i 1 u S i 1 ½ D R c (u) = ( u 2 + c(x)u 2 )dx, D u2 dx ß S 1 Ð W 1,2 0 (D) i -. (W 1,2 0 (D) ÙÌ Ó) c(x) 0, ½ (3.16) Æ «, ÄÆ Ô Ý (λ i, ϕ i ). (3.15) Fourier ¾ u(t, x) = c i e (a dλ1)t ϕ i (x), (3.18) i=1

47 44 Þ c i = D u 0(x)ϕ i (x)dx, D ϕ2 i (x) = 1, D ϕ i(x)ϕ j (x)dx = 0(i j). (3.18) Ð, a < dλ 1, Ã t, u(t, x) 0, x D; a > dλ 1, u(t, x) Íº Æ. Ø a = dλ 1 Đ (3.15) ÆÐ. a Í Ç, Ð Đ, ½ D Þ Ë ½ßÎ. Ü 3.1-(3) Ð D 1 D 2, Ã λ 1 (D 1 ) > λ 2 (D 2 ). n = 1, D = (0, lπ), (3.16) V = λv, x (0, lπ), V (0) = V (lπ) = 0, (3.19) Ô λ i (l) = i 2 /l 2, Ý ϕ i (x) = sin(ix/l). l ÇĐ, λ i (l) Î. µ Ç, Æ¼º Ç ÝÆ Ð. ¼Ó Ä Ñ ¹ (habitat fragmentation). Ó Đ t D t ½ D t Þ Î. Æ Đ D t Đ λ 1 (D t ) < a/d, u(t, x) ÊÝ (t 0), Ü Ð t = t 0, λ 1 (D t0 ) = a/d t > t 0 Å λ 1 (D t ) > a/d ÃØ¼ÐÆÊÝ, ½Å (3.15) Þ± Ç. Á λ 1 (D) = a/d Æ D Î ß (minimal patch size). Ð ÔĐÎĐ D Æ. Đ (½ ÍÌ ), Ã Î ß º ÆÒ É Ð (3.19) Æ Î ß Ò d l = a. Ô Ð ÆÍ Î ß ÆÉ Đ ÂÆÒ ÂÍ Neumann Â (ºÅ³ no flux) Ã ¾ D ØÐ ß Ô u = u n = 0. (3.20) n d V = λv, x D, V (3.21) n = 0, x D Æ Ð Ô 0 Ð Ý C 0 Â (3.20) ¼Ð ÆÁ ÓÐ ÆÇ a Å Ð ³ Õ Ç ½ u t = d u + au, t > 0, x D, u(0, x) = u 0 (x), x D, u (t, x) = 0, t > 0, x D, n Æ u(t, x) D ³ d u udx = dt D D t dx = (d u + au)dx = d D D D u(t, x)dx = e at u n ds + a udx = a udx, D D D u 0 (x)dx. (3.22)

48 Dirichlet Ô Â ½ (3.15) ÞÆ Â Á x u < 0 a Đ ¼Ð Ê n Ý ¾Ð Æ Ô ÂÍ Robin Â u = au, x D. (3.23) n Ð ÂÆÛ«Í Ð ÆÐ D Æ» ÆÅ± ½ Dirichlet Ô ( Â ÆÐ) Neumann Ô (Ü ÆÐ) Ñ ¹Ð Âß Ð S u + (1 S)u = 0, x D. (3.24) n 0 S 1 Ð Â² µ Dirichlet(S = 0) Neumann(S = 1) Robin(0 < S < 1) ¹ Â ÄÕÐ Robin Â Î ß ÆÉ Í± «Æ Ù Ü (n = 1 «) ± 3.4 ²±ÏÑ Æ 45 (3.1) Æ Þ Ä ¼ÜÆ³ÆÈ Ü ÐµÁ ÍÐ ¼ Ø Ä ß ¼Ü ¾ ÂÎ Í ÆØ» u (t, x)dx + div(j(t, x))dx f(t, x, u)dx = 0, t ½» O O u (t, x) + div(j(t, x))dx = f(t, x, u). (3.25) t f(t, x, u) t Ú x Å± u Æ Â ÆÇ ¼ÜÆÎ Í ÍÐ Ó ( ) ¼ (Ð Ô ) Ø Fick Ã (3.25) Ð Æ» (reaction-diffusion equation) ÐÍÞÆ Â Ý Æ» ½ ÐÍ Ä ÌÆ Fisher-KPP ½ Logistic Æ» O u (t, x) = d u(t, x) + f(t, x, u). (3.26) t u (t, x) = d u(t, x) + f(u). (3.27) t u t = d u + au(1 u ). (3.28) N À 1937 ¼ ÕÓ Fisher(Fisher,1937) Ð Ø Ì Ó ½ ÐµÐ ĐÞÆ Ó ± (Kolmogoroff, Petrovsky, Piscounoff, 1937) Fisher Æ ÔÍ 2.6 ÏÞÆ Fisher- Haldane-Wright Í u(t, x) ÆÛ«Í Ú x ÔÆ Æ½ Ð Æ ÍÐ ÆÝ Ã Ä D = R 1 N = 1 Æ «u t = du xx + au(1 u), t > 0, x R. (3.29) Æ (traveling wave) : u(t, x) = v(x ct) c Ú v(z) cv = dv + av(1 v),

49 46 v = w, w = c d w a v(1 v). (3.30) d Á ½ (3.30) ±± Ü (0, 0), (1, 0) ± ÔÆ Jacobian Ö J(0, 0) = 0 1 a c, J(1, 0) = a 0 1 c d d d d Ø (1, 0) ÍÐ ½ (1, 0) Ô Tr(J(1, 0)) = c d < 0, Det(J(1, 0)) = a > 0 (0, 0) Íµ d Æ Í T 2 4D = c2 4ad d 2 Ä c 4ad (0, 0) ÍÐ µ ½ c < 4ad (0, 0) ÍÐ µ Ð ¾ß Â À: a = d = 1, ( ) c = 2.1 Â ; ( µ) c = 0.3 Â ; ( ) c = 2.1 Æ; ( µ) c = 0.3 Æ. ÜÁ ½ Ð c > 0 (1, 0) Æ µ Õ Ð Å (0, 0) ½ ÍÐ Ì (0, 0) (1, 0) Æ ÛÕ (0, 0) ÍÐ µ Ð ÐÕ Æ v- Ô ÍÄÆ Á v Æ¼ «½ c 4ad v- Ô Ý Ä Åµ 3.2. c 4ad µù (3.29) Ù ÍÍ u(t, x) = v(x ct) Á 0 < v(z) < 1, v (z) < 0, lim v(z) = 1, lim z v(z) = 0. z (i) Ð Ò Ç Ð Fisher-KPP Ð Ç ÐÆ : c = 5 1 v(z) = 6 (1 + k exp( z, k Í (Ablowitz, Zeppetella, 1979.) )) 2 6

50 (ii) Æ Þ Ä c > 0 Ü½ Å Æ Ü É Õ c < 0 Ã Ä Å Ç Ü É Õ (iii) c = 4ad ÎÚ (minimal wave speed) 3.2 ÏÞ Æ ÐÙ ÌÁ (Kolmogoroff, Petrovsky, Piscounoff,, 1937) Ëµ Ô u 0 (x) = 1, x 0, u 0 (x) = 0, x > 0 (3.29) Æ Ð Ú± 4ad Æ ½ ÎÚÍ¼ÜÕ Æ ÁÚ± 3.5 ÎÌ»Ð 1. u t = du xx Æ u(t, x) = v( x t ) Æ ÜÞ ÐÆ» Æ» 2.»Ø Ü n = 0, ±Ð x = 0 Ô ÁÐ Å È È¼ ÆÉ Þ p Ô ÆÉ 1 2p n ÁÅ x = m ÆÉ º n m n ÐÉ À»Æ» Æ» ± u t = u xx + au, 0 < x < lπ, t > 0, u(0, x) = u 0 (x), 0 x lπ,. ku(t, 0) (1 k)u x (t, 0) = 0, ku(t, lπ) + (1 k)u x (t, lπ) = 0. u(t, x) Üº Å 0 Æ a = a 0 Ð Î ß ÆĐÎ l 4. Æ Ô Ý 5. Allen-Cahn y + 2y + ky = 0, 0 < x < π, y(0) = y(π) = 0. u t = du xx + u(1 u)(1 + u), t > 0, x R, ±Ð Ú 0 Æ ß Å (standing wave) d = 1 ÆÅ ½ u(x) u + u u 3 = 0, x R, u(± ) = ±1. 6. Ù± Allee Ï Æ Æ» u t = du xx + u(1 u)(u a), t > 0, x R, 0 < a < 1 ½ Æß Ð ÐÚ c

51 48 7. Æ» u t = du xx, t > 0, x (0, ), v t = dv xx, t > 0, x (0, ), w t = dw xx, t > 0, x (0, ), u(t, 0) = v(t, 0) = w(t, 0), t > 0, u x (t, 0) = v x (t, 0) + w x (t, 0), t > 0, u(0, x) = u 0 (x), v(0, x) = v 0 (x), w(0, x) = w 0 (x), x (0, ). Æ ½ ( Ð )

52 Í ± ÎÐ ÉÔÅµ 4.1 Banach Ó Ä Ð «. X ÍÐ (linear space) ½ x, y X, ± x + y X; k R, x X, ± kx X, ß Ð Ý : X R Å (i) x 0, x = 0 x = 0; (ii) a x = a x ; (iii) x + y x + y ; Ã (X, ) Ð Â (normed linear space). x x Æ (norm). Ð Â ¹Æ (complete), ÞÆ Cauchy ± Ð ¹Æ Â Banach (Banach space). 1. X, Y Í± Banach g : X Y ÍÐ (map), g º Æ (linear), (i) x, y X, g(x + y) = g(x) + g(y); (ii) k R, x X, g(kx) = kg(x). Ü X Å Y Æ L(X, Y ); 2. x 0 X, g x 0 Í Æ (continuous) ε > 0, ß δ > 0, x x 0 X < δ, ± g(x) g(x 0 ) Y < ε; 3. g x 0 Í Fréchet Æ (Fréchet differentiable), ß A L(X, Y ), ε > 0, ß δ > 0, Å x x 0 X < δ, ± g(x) g(x 0 ) A(x x 0 ) Y < ε. X Y Í X Y Æ. ¾Ð «Í: g x 0 Í Gáteaux (Gáteaux differentiable), h X, g(x 0 + th) g(x) ta(h) Y lim t 0 t = 0. g Í Fréchet Æ, Í Gáteaux Æ. À Þ, ± Đ. º ³ Å Ý ½ÊÃ. 4.2 Ü ÀÆ Banach Í R n, n 1. ± Banach, Ð Í Æ. Ä Ø ÒÞ ÆÝ Í Banach, : Ω Í R n ÞÐ ± ¾ Ù±Ð, 1. C 0 ( Ω) = {f : Ω R, f Ω ÁÐ }; 49

53 50 2. C k ( Ω) = {f : Ω R, D i f C 0 ( Ω), 0 i k}; k 1, k N k =, D i f Í f Æ i Í Ø. C k ( Ω) Æ Í f Ck ( ω)= max D i f(x). x Ω 0 i k 3. C k,α ( Ω) = {f C k ( Ω), sup x,y Ω,x =y D k f(x) D k f(y) x y α < }, C k,α ( Ω) Æ Í f C k,α ( Ω)= f Ck ( Ω) + C k,α ( Ω) Hölder (Hölder space). sup x,y Ω,x =y D k f(x) D k f(y) x y α, u = f (4.1) Æ, Ä± Schauder ; f C α ( Ω), (4.1) ± Ð u C 2,α ( Ω) C Đ Ω. u C 2,α ( Ω) C f C α ( Ω), (4.2) f C α ( Ω), «supp f = {x Ω : f(x) 0} f Æ (support). supp f Ω, Ã f Ù± (compact support). 0 k, 0 α < 1, C k,α 0 ( Ω) = {f C k,α ( Ω) : supp(d i f) Ω, 0 i k} 4. L p (Ω) = {f Lebesgue ³, Ω f(x) p dx < }; L (Ω) = {f Lebesgue ³, f(x) < ÒÐ ½É }; : f Lp (Ω)= ( Ω ) 1 f(x) p p dx ; f L (Ω) = essmax x Ω f(x). u L 1 (Ω), Ä «u Æ (weak derivative), u, v L 1 (Ω), φ C 0 (Ω), ± Ã v Í u x i Æ Ø, v = u x i. ÍÍ «. Ω u φ dx = vφdx, x i Ω W k,p (Ω) = {f L p (Ω) : D i f L p (Ω), 0 i k}, f W k,p (Ω)= 0 i k D i f L p (Ω). Ë C k,α ( Ω) Í C (Ω) C k,α ( Ω) Æ ², W k,p (Ω) Í C W k,p (Ω) Æ ². Äß «C k,α ( Ω) W k,p (Ω) Æ ; C k,α 0 ( Ω) Í C0 (Ω) C k,α ( Ω) Æ ² W k,p 0 (Ω) Í W k,p W k,p (Ω) Æ ².

54 W k,p (Ω), W k,p 0 (Ω) Í Sobolev ( Sobolev space). (4.1) Æ ß Sobolev Å, ½ L p. p 2, f L p (Ω), ß ÐÆ u W 2,p (Ω) (4.1) u W 2,p (Ω) C f L p (Ω), (4.3) 51 Dirichlet Ô u = f, x Ω, u = 0, x Ω. (4.4) «ÝÆ Hölder L p ß, f C α ( Ω), Ã u C 2,α 0 ( Ω); f L p ( Ω), Ã u W 2,p (Ω) W 2,p 0 (Ω). Å Ä± Sobolev µ : 1. kp < n, W k,p (Ω) µå L q (Ω), 1 q np n kp ; 2. kp = n, W k,p (Ω) µå L q (Ω), 1 q < ; 3. kp > n, W k,p (Ω) µå C α ( Ω), 1 q np n kp, Þ α = k n p, k n p < 1; α [0, 1), k n p = 1; α = 1, k n p > 1. Ð À Í Rellich W 1,2 0 (Ω) µå L 2 (Ω). Äß : A L(X, Y ), ± B X, A(B) Í Æ, A Ü X Å Y Æ Ü ; Ð «: A L(X, Y ), ± X ÞÆ {x n }, {A(x n )} ± Y ÞÆ. Ä F : R X Y, X C 2,α 0 (Ω) W 2,p (Ω) W 1,p 0 (Ω), Y C α ( Ω) L p (Ω), «Ã F ÆÐÚ ÚØ F(λ, u) = u + λf(u), (4.5) F λ (λ, u)[τ] = τf(u), F u (λ, u)[w] = w + λf (u)w, F λλ [τ 1, τ 2 ] = 0, F λu [τ, w] = τf (u)w, F uu [w 1, w 2 ] = λf (u)w 1 w 2. g : X Y, Í Í L(X, L(X, Y )) ÆÐ, ½ Í Đ X ÆÐ. L(X, L(X, Y )) = L(X, X, Y ) 4.3 Ü ÕÆ ±µ Æ ¹ÐÂ, Ä Banach ÞÆ Ý.

55 X, Y, Z µ Banach, U Z X Ð (λ 0, u 0 ) Í ºÒ. Õ F : U Y ÐÍ ±Ã ³ Ô, F(λ 0, u 0 ) = 0 Ë F u (λ 0, u 0 ) ÐÍ Á Ô, F u (λ 0, u 0 ) Ð ÔÅ Ô, Ë Fu 1(λ 0, u 0 ) : Y X Ð Æ. Ú Ù λ 0 Ù Z Í ºÒ A, u 0 Ù X Í ºÒ B, Ü Æ λ A Ù Í u(λ) B F(λ, u(λ)) = 0 Ë u( ) : A B Ð±Ã ³, u (λ 0 ) : X Y µ u (λ 0 )[ψ] = [F u (λ 0, u 0 )] 1 F λ (λ 0, u 0 )[ψ]. u u (λ(s),u(s)) (λ, u(λ)) u 0 u 0 λ 0 λ λ 0 λ 4.1 Æ À ( ) Þ. (µ). Ý ÒËÐ Æ Í Æ (½ Ü Í Æ), Z = R, Ã Æ {(λ, u)} ÍÐ, º λ Å. Ä Ç F u (λ 0, u 0 ) Æ «. Í F(λ, u) = 0, λ R, u X, (4.6) F : R X Y, X Y Í Banach. Ä F(λ 0, u 0 ) = 0, (F1) dimn(f u (λ 0, u 0 )) = codimr(f u (λ 0, u 0 )) = 1, N(F u (λ 0, u 0 )) = span{w 0 }. N(L) R(L) Í L : X Y Æ½ Ô¾ R(F u (λ 0, u 0 )) º 1, Ãß Y Æ Ý l Y (Y Æ ) Å u R(F u (λ 0, u 0 )) l, u = 0, l, u Í Y Y Æ. Ä± 4.2. (» ÏÓÄ Saddle-Node Bifurcation Theorem) U Ð (λ 0, u 0 ) Ù R X Í ºÒ, F : U Y Ð±Ã ³ Ô. F(λ 0, u 0 ) = 0, F Ù (λ 0, u 0 ) (F1) Å (F2) F λ (λ 0, u 0 ) R(F u (λ 0, u 0 )). Ú (1) Z Ð span{w 0 } Ù X Í (4.6) Ù (λ 0, u 0 ) ¾ Í {(λ(s), u(s)) = (λ(s), u 0 + sw 0 + z(s)) : s < δ}, Æ s (λ(s), z(s)) R Z Ð±Ã ³, λ(0) = u 0, λ (0) = 0, z(0) = z (0) = 0.

56 53 (2) F Ù (λ 0, u 0 ) ¾ u ³, Ú λ(s) Ù λ = λ 0 ³, Ë λ (0) = l, F uu(λ 0, u 0 )[w 0, w 0 ]. l, F λ (λ 0, u 0 )»Ü ÉÉÆ Ý, Í, Í ÞÆ º λ Å. ÌÁ λ (0) 0, Ã λ 0 ÆÐ È¼¾, (4.6) (λ 0, u 0 ) Å Ð±±. F (F4 ) F λ (λ 0, u 0 ) R(F u (λ 0, u 0 )). λ (0) = 0, F (λ 0, u 0 ) Å ±Ð. ÐÁ Æ Â, 1.6 Æ. Å Ä± Æ ( ÓÄ Transcritical or Pitchfork Bifurcation Theorem) U Ð (λ 0, u 0 ) R X Í ºÒ, F : U Y ÐÍ ±Ã ³ Ô, Æ (λ, u 0 ) U, F(λ, u 0 ) = 0. Ù (λ 0, u 0 ), F (F1) Å (F3) F λu (λ 0, u 0 )[w 0 ] R(F u (λ 0, u 0 )). Õ Z Ð span{w 0 } Ù X Í, Ú (4.6) Ù (λ 0, u 0 ) ¾ ÉÃµ³ : u = u 0 Å {(λ(s), u(s)) = (λ(s), u 0 + sw 0 + z(s) : s I = ( ε, ε)}, λ (λ(s), z(s)) R Z Ð±Ã ³, Ü λ(0) = λ 0, z(0) = z (0) = 0 Ë λ (0) = l, F uu(λ 0, u 0 )[w 0, w 0 ] 2 l, F λu (λ 0, u 0 )[w 0 ]. (4.7) (F4 ), Ú λ (0) = 0. ßÙ F Ù (λ 0, u 0 ) ¾ ³, Ú λ(s) µ ³, Ë Æ θ λ (0) = l, F uuu(λ 0, u 0 )[w 0, w 0, w 0 ] + 3 l, F uu (λ 0, u 0 )[w 0, θ], 3 l, F λu (λ 0, u 0 )[w 0 ] F uu (λ 0, u 0 )[w 0, w 0 ] + F u (λ 0, u 0 )[θ] = Þ λ (0) 0, Ð ; λ (0) = 0 ½ λ (0) 0, Ð Ê. Í± «ÑÔ ±Ð Ü»Ð Ü ÁÈÊ Ðµ 1.2 Ï ËÆ É. ± Æ Ð Ç 70 (Crandall- Rabinowitz, 1971,1972) ÞÐ, Ë ÅÀ ³ (Shi,2009). Ä ÈĐ Ä Æ» ( ) ÆÜ Æ. 4.4 Æ» ²±ÏÑ ÊÕÆ Ä ÇÆ Fisher-KPP Ñ ÇÆÜ : u + c(x)u = au uf(u), x Ω, u = 0, x Ω, (4.8)

57 54 y y λ λ 4.2 Æ ¹ ( ). (µ) Ë. c(x) C 0 ( Ω), a > 0, f(u) C 2 (R + ) f(0) = 0, f (u) 0, lim f(u) =. Ð Æ u u = f(x, u), x Ω, u = 0, x Ω, u Í (4.9) Æ f u (x, u) C 0 ( Ω), Ã u (4.9) Æµ ÇÆ Ü φ f u (x, u)φ = µφ, x Ω, (4.10) φ = 0, x Ω. 3.1, (4.10) ±Ð Ô µ i (u), µ i (u) µ i+1 (u), lim i µ i (u) =. µ 1 (u) > 0, Ä u Í (4.9) Æµ (stable), Ã u Í µ Æ (unstable). ÄÌ Þ 4.4. (4.8), a 0 = λ c µ + c(x) (Ú 3.1) Ú (4.9) (1) «a a 0 Ù (4.8) Í ¼ µ u = 0; (2) «a > a 0 Ù (4.8) Í u a (x) Þ u = 0»Æ ¼ Å {(a, u a ) : a > a 0 } R C 2,α 0 ( Ω) Ð±Ã ³ Ë u a (x) a Æ Û u a (x) Ð (4.8) u (λ,u a ) a0 a 4.3 (4.8) Ê

58 4.4 ± º «Æ Ë (Brezis-Kamin, 1992) ÄÅ Ý ÆÅ ÐÁ: a a 0 (4.8) Æ Ä Ð φ + c(x)φ = λφ, x Ω, φ = 0, x Ω, 55 (4.11) ± Î Ô λ c 1(= a 0 ) R, Á Æ Ý φ 1 (x) > 0, x Ω. u Í (4.8) Æ Ä Æ (4.8) φ 1 Ω ³ «Æ φ 1 Æ u Ω ³ Ω φ 1 + c(x)φ 1 = a 0 φ 1, x Ω, φ 1 = 0, Ω x Ω. ±³ Á Ø Green Ç ( u φ 1 φ 1 u)dx = ( u n φ 1 φ 1 u)dx = 0, n a uφ 1 dx uf(u)φ 1 dx = a 0 uφ 1 dx. Ω Ω Ω f(u) 0, u 0, φ 1 > 0, a a 0, a = a 0 u 0. Á a = a 0 ÍÐ Ç u = 0 Í (4.8) ÆÐ Ü Ä «F : R X Y, u X C 2,α 0 ( Ω), F Í Ú Æ (4.12) F(a, u) = u c(x)u + au uf(u), (4.13) F u (a, 0)[φ] = φ c(x)φ + aφ, φ X. (4.14) a = a 0 F u (a, 0)[φ] = 0 ± Ð φ 1 > 0 Å Ø dimn ( F u (a, 0) ) = 1, spann ( F u (a, 0) ) = {φ 1 }, h R ( F u (a, 0) ), Ãß v X Å v c(x)v + a 0 v = h, (4.15) (4.15) Æ±² φ 1 ³ Ω hφ 1dx = 0. ¾Ð Ç Fredholm Ð Ω hφ 1dx = 0, Ã (4.15) ± v. R ( F u (a, 0) ) = {h C α ( Ω) : Ω hφ 1dx = 0}, Ø F (F1). Å ÄÛ (F3), F λu (a 0, 0)[φ 1 ] R ( F u (a, 0) ), Ω φ 1 φ 1 dx 0, Ä 4.3 F(a, u) = 0 (a 0, 0) Å Æ {(a, 0) : λ R} { ( a(s), u(s) ) : s < δ} Å a(0) = a 0, u(s) = sφ 1 + sz(s), z(0) = 0, a Ω (0) = f (0)φ 3 1 dx 2 Ω φ2 1 dx > ÞÆ Ý l, ψ = Ω φ 1ψdx, ½ F uu (a 0, 0)[ψ 1, ψ 2 ] = f (0)ψ 1 ψ 2. Ý 0 < s < δ a(s) > a 0, ½ u(s) = sφ 1 + sz(s) > 0, z(0) = 0 Þ ÎÐ Æ δå ØÐ Ç a = a 0 s (0, δ) ( λ(s) > a 0 ), u(s) Í s ( δ, 0) ( λ(s) < a0 ), u(s) ÍÄ ¹Á (4.8) Æ Íµ Æ

59 (a, u) = (a 0,0) Æ Ä u Í (4.8) Æ Ä Ã Ω u 0 u > 0 Ô Ë (À» Í) u Í (4.8) Æ Ã u Í ψ + c(x)ψ 1 = aψ f(u)ψ + µψ, x Ω, (4.16) φ = 0, x Ω ÆÐ Ý Å µ = 0. Ð ÔÍ ÐÙ± Ý Æ Ô (4.16) Æ Ð Ô µ 1 = 0. (4.8) Æ u Í (4.8) ÆÐ ψ + c(x)ψ = aψ f(u)ψ f (u)uψ + µψ, x Ω, (4.17) φ = 0, x Ω, 3.1, (4.17) Æ Ð Ô µ µ 1 = min Ø u Íµ Æ ψ W 1,2 0 (Ω), ψ 0 min ψ W 1,2 0 (Ω), ψ 0 = µ 1 = 0, 1 Ω ( ψ 2 + c(x)ψ 2 aψ 2 + f(u)ψ 2 + f (u)uψ 2) dx Ω ψ2 dx ( Ω ψ 2 + c(x)ψ 2 aψ 2 + f(u)ψ 2) dx, Ω ψ2 dx ÕÁ a 1 > a 0, ± (4.8) Æ a [a 0, a 1 ] ÍÐÙ± Æ u u = u ( a c(x) f(u) ), x Ω. lim f(u) = a [a 0, a 1 ], ß u u 1 > 0 Å u > u 1 f(u) > a 1 + c(x) C ( Ω). a [a 0 0, a 1 ] u 0 < u(x) u 1. Ã u(x) = u(x 1 ) > u 1, Ã u(x 1 ) < 0,» u x 1 ÔÐÅ ĐÔ ¹ max x Ω»Á Ë ÁÐ a (a 0, a 0 +δ) (4.8) ±Ð Ü 0 Î ¹ÁÐ Íµ Æ Ø ½ F u (a, u) Í Æ Ø (a, u) Å Ø 4.1 ÍÐ Ð

60 «Σ (4.8) Æ ½ Σ = {(a, u) : a > a 0, (a, u)í(4.8)æ }. Σ 1 Σ Æ²Û ÁÞ Æ Å «a 2 = sup{a > a 0 : ß (4.8)Æ (a, u) Σ 1 }. Ã Σ 1 = {(a, u a ) : a 0 < a < a 2 } Í Ð Ð Σ 1 ÕÂ ÁÐ (a, u a ) ÍÐ Ä Ë u a a Ç ÌÁ (4.8) Þ a u a u a a = 0, a + c(x) u a a = a u a a f(u a) u a a f u a (u a )u a a + u a = 0, x Ω, x Ω, ÆÒ L( u a a ) = u a, L Í (4.17) ÞÅ Lψ = µψ Æ Ü ¹ÁÐ L Æ Ð Ô µ 1 > 0, Ã ÅÇ ËÆ Ô Ð u a a > 0, u a > 0. a 2 <, lim u a (x) = u (x) ß a a 2 ÕÁ u (x) Í± Æ ± ÇÆ Schauder Ë u C 2,α 0 ( Ω) (a 2, u ) Í a = a 2 ÆÐ u > 0. Ý 4.1Å, (a 2, u ) ß Σ 1 º Ø Ø a 2 + δ 2 > a 2.» a 2 Æ «¹ Ø a 2 =. Å (4.8) ±Òµ Σ 1 Æ (a, u a ), a > a 0, ² Ý Ë (a, u a ) Ð Ð Σ 2, Σ 2 ºØ Ø a (a 0, ), (4.8) a = a 0 Ô ± 0 Σ 2» Σ 1 Đ Σ = Σ 1 = {(a, u a ) : a > a 0 } ÍÐ Ð ØØ 4.4 ¾ Á (1) µ Ë Ä f C 2 ( R + ). Á ËÞ Ä Þ f u = 0 Ú ½ Å 4.3. ÄÅ Æ 4.3 ß½ (Crandall-Rabinowitz, 1971) Æ» Â Å F ÍÐÚ F λu (λ 0, u 0 ) ß ½ Í Ý Ð Ü Í C 1 Æ ßÔÀ± (4.7). Å ÂË Æ f C 1 ( R + ), Ä² Ë 4.4 Òµ a = a 0 ÆÐ Å Ý ÆÅ Þ f C 1 ( R + ). Ð Â Σ 1 Ò a = a 0 Ð ²ÍÐ Æ Å Ø f C α ( R + ), 0 < α < 1, Ä² Å«ÝÆ Σ 1 R C 2,β 0 ( Ω), 0 < β < α. ÍØ Σ 1 Æß Þ Ë Ðß Þ Ð (2) 4.4 Æ Ñ Ð À 4.5 ÏÞ Ð Î ÆÜ Æ ¾Ð Fisher-KPP u + au uf(u) = 0, x Ω, u = 0, x Ω, 57 u + λug(u) = 0, x Ω, u = 0, x Ω, (4.18) ug(u) ÍÐ 1.4 ÏÞ «ÆÑ«Logistic Ç ÎÇ ½ g(0) > 0, g(u) ß u 0 > 0 Å g(u 0 ) = 0 g(u) > 0, u (0, ) ½ lim g(u) = 0. g(u) = 1 u(logistic), u b g(u) = 1 + (1 + au) p, (a, p > 0, b > 1) (Hassell, Beventon-Holt), g(u) = 1 + ae u, (a > 1) (Ricker, Nicholson s blowflies), g(u) = 1 u, (c > 0) ( ¼± food-limited) (4.18)» (4.8) 1 + cu Å λ ÜÊ Æ 4.4 Æ Ë Å (4.18), ± ÎÈ ½ Ù Ï ±