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2 Malthus logistic Lotoka-Volterra Lotka-Volterra Lotka-Volterra Lotka-Volterra Z

3 1 [1] 1.1 Malthus N 0 R N t 1 N t N t = RN t 1 N t = R t N 0 t N t N(t) r 2

4 ( )0 N 0 t N(t) r, dn(t) = rn(t) N(t) = N 0 e rt N(t) r (Malthusian parameter) 1.1: Malthus 1.2 logistic K K (carrying capacity) dn(t) ( = r 1 N ) N K 3

5 logistic (logistic equation) logistic N(t) = K 1 + (K N 0 )/N 0 e rt 1.2: logistic N(t) K r N(t) N(t) K N(t) K 4

6 2 [2],Hirsch Smale Devaney [3] (Volterra) 2.1 Lotoka-Volterra x y y = 0 = dx(t) = ax (a > 0) x(t) = x 0 e at bxy (y > 0) dx(t) = ax bxy x = 0 = dy(t) = cy (c > 0) y t = y 0 e ct 5

7 dxy dy(t) = cx + dxy dx(t) = (a by)x dy(t) = ( c + dx)y (Lotka) (Volterra) Lotka Volterra a, b, c, d x, y Lotka-Volterra (x, y) = (c/d, a/b) ( ) X a bx bx = dx c + dx a c y x (c/d, a/b) ±i ac 2.3 Lotka-Volterra x x = 0 y = a/b y y = 0 x = c/d 0 x, y > 0 L L(x, y) = F (x, y) + G(x, y) 6

8 L L L 0 L(x, y) = x df dx L(x, y) = d L(x(t), y(t)) = df dx x + dg dy y dg (a by) + y ( c + dx) dy x df/dx dx c y dg/dy by a x y 1 x df/dx dx c = y dg/dy by a =. df dx = d c x dg dy = b a y F (x) = dx c log x G(y) = dy a log y L(x, y) = dx c log x + by a log y x,y> 0 L/ x L/ y Z = (c/d, a/b) L L L L(z)) Z 2.4 Lotka-VolterraZ 7

9 W Z W x y Z n ± t n ± < t 1 < t 0 < t 1 < ϕ tn (W ) x = c/d W ϕ tn W ) x = c/d n ϕ tn (W ) Z n ϕ tn (W ) Z L W L(W ) = L(Z) L(Z) Z 2.1: Lotka-Volterra 8

10 3 [1] 3.1 (species replacement) 9

11 3.2 N 1 (t) N 2 (t) N 3 (t) dn 1 (t) = c 1 + [r 1 a 11 N 1 a 12 N 2 a 13 N 3 ]N 1 dn 2 (t) = c 2 + [r 2 a 21 N 1 a 22 N 2 a 23 N 3 ]N 2 dn 3 (t) = c 3 + [r 3 a 31 N 1 a 32 N 2 a 33 N 3 ]N 3 c i r i a ij j i a 11 a 22 > a 12 a 21 a 11 a 22 < a 12 a 21 N 2 = N 3 = 0 logistic N 3 = 0 Lotka-Volterra = 0 c i = 0 10

12 (N 1,N 2,N 3 ) N 1 = (a 23a 32 r 1 a 22 a 33 r 1 a 13 a 32 r 2 + a 12 a 33 r 2 + a 13 a 22 r 3 a 12 a 23 r 3 ) D N 2 = (a 23a 31 r 1 a 21 a 33 r 1 a 13 a 31 r 2 + a 11 a 33 r 2 + a 13 a 21 r 3 a 11 a 23 r 3 ) D N 3 = (a 22a 31 r 1 a 21 a 32 r 1 a 12 a 31 r 2 + a 11 a 32 r 2 + a 12 a 21 r 3 a 11 a 22 r 3 ) D D = ( a 13 a 22 a 31 + a 12 a 23 a 31 + a 13 a 21 a 32 a 11 a 23 a 32 a 12 a 21 a 33 + a 11 a 22 a 33 ) N 1 r 1 /a 11 dn 2 (t)/ r 2 a 21 N 1 > 0 N 1 r 1 /a 11 a 11 r 2 > a 21 r 1 (3.1) a 11 r 3 > a 31 r 1 (3.2) N 2 r 2 /a 22 a 22 r 3 > a 32 r 2 (3.3) a 22 r 1 > a 12 r 2 (3.4) 11

13 a 33 r 1 > a 13 r 3 (3.5) a 33 r 2 > a 12 r 3 (3.6) 3.3 c 1, c 2, c 3 = : c i = 0 3 c i = 0 limn i (t) = 0 (saddle connection) heteroclinic c i (limit cycle) 12

14 (1),(2) : (1),(2) : 3 (a 11, a 12, a 13, a 21, a 22, a 23, a 31, a 32, a 33 ) = (0.2, 0.4, 0.1, 0.1, 0.2, 0.4, 0.4, 0.1, 0.2) c i = 0.1 (r 1, r 2, r 3 ) = (0.5, 0.6, 0.7) 13

15 [1],, (2000). [2],, (1997). [3] Hirsch Smale Devaney,, (2007). [4],, (2004). [5],, (1997). [6],,, (2010). [7],,, (1998). [8],,, (1992). [9],,, (2002). [10] Michael E Gilpin, Limit Cycles in Competition Communities, The American Naturalist 109 (1975). [11] Hiroyuki MATSUDA Tokio WADA Yasuhiro TAKEUCHI Yoshiharu MAT- SUMIYA,MODEL ANALYSIS OF THE EFFECT OF ENVIRONMENTAL FLUC- TUATION ON THE SPECIES REPLACEMENT PATTERN OF PELAGIC FISHES UNDER INTERSPECIFIC COMPETITION,Res Popul Ecol (1992). 14

1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0

1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx