8 2 次元の微分方程式（Lotka-Volterra...
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53 8 2 次元の微分方程式（Lotka-Volterra の競争モデル） 8.1 ベクトル場の解析（相平面解析） 前回、次のような共通の資源を巡って競争関係にある 2 種類の個体群動態を表す Lotka-Volterra の競争モデルを紹介した。 dx(t) dt = rx (t) (1 - αx(t) - y(t)) (8.1a) dy(t) dt = ry (t) (1 - x(t) - βy(t)) (8.1b) ここで、α > 0, β > 0 は種内の競争率を表すパラメータ、r> 0 は各個体群の成長率 を表している。 個体群の均衡状態 dx(t) dt =0, dy(t) dt =0 を表す解が (8.1) の平衡点である。(8.1) の平衡点を (x * ,y * ) で表すと、(x * ,y * ) は以 下の式を満たす。 0= rx * (1 - αx * - y * ) , (8.2a) 0= ry * (1 - x * - βy * ) . (8.2b) (8.2) から Lotka-Volterra 競争モデル (8.1) の平衡点は以下のようにまとめられる。 平衡点 存在条件 (0, 0) - 2 種の個体群が共に絶滅 1 α , 0 - 個体群 x のみが生存 0, 1 β - 個体群 y のみが生存 β - 1 αβ - 1 , α - 1 αβ - 1 α, β < 1 または α, β > 1 2 種の個体群 x, y が共存 演習 8.1. αβ =1 のとき、2 種の個体群 x, y が共存する平衡点が存在するか調べよ。 前回の講義で、以下のように定めた微分方程式 (8.1) の右辺 f (x, y)= f (x, y) g(x, y) = rx (1 - αx - y) ry (1 - x - βy)
Transcript of 8 2 次元の微分方程式（Lotka-Volterra...
Lotka-Volterra
dx(t)
dy(t)
! > 0, " > 0r > 0
dx(t)
0 = rx! (1! !x! ! y!) ,(8.2a)
0 = ry! (1! x! ! "y!) .(8.2b)
(8.2) Lotka-Volterra (8.1) (0, 0) - 2! 1
! , 0
8.1. !" = 12 x, y (8.1)
f(x, y) =
(8.3)
%
&
f(x, y), g(x, y)x(t), y(t) 8.2. xy 1f(x, y), g(x, y) f(x, y) g(x, y) f(x, y) = 0, g(x, y) = 0 (x, y)
f(x, y) = 0 "# x = 0 1! !x! y = 0,
g(x, y) = 0 "# y = 0 1! x! "y = 0.
#1 : 1! !x! y = 0(8.4)
#2 : 1! x! "y = 0(8.5)
#1, #2 xyf(x, y) = 0, g(x, y) = 0 (x, y) 8.1! > 1, " > 1 #1, #2 8.3. ! < 1, " < 1 #1, #2 xy ! < 1 < "
#1, #2 xy ! > 1, " > 1 8.1xy
1 #1, #2 4
D1 = {(x, y) : 1! !x! y > 0, 1! x! "y > 0}
D2 = {(x, y) : 1! !x! y < 0, 1! x! "y > 0}
D3 = {(x, y) : 1! !x! y < 0, 1! x! "y < 0}
D4 = {(x, y) : 1! !x! y > 0, 1! x! "y < 0} .
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8.1. xy (8.1) #1, #2 Dj (j = 1, 2, 3, 4) . x, y
Dj , j = 1, 2, 3, 4f(x, y), g(x, y) f(x, y) > 0, g(x, y) > 0, (x, y) $ D1
f(x, y) > 0, g(x, y) < 0, (x, y) $ D2
f(x, y) < 0, g(x, y) < 0, (x, y) $ D3
f(x, y) < 0, g(x, y) > 0, (x, y) $ D4
(8.3)xyDj (j = 1, 2, 3, 4) x, y 8.1 8.1x, y 2 (8.6) (x!, y!) =
! " ! 1
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8.1. Lotka-Volterra (8.1)! > 1, " > 12 (x!, y!) 2 8.1. ! < 1, " < 1
8.2
(8.1) 8.1 2 (x!, y!)
8.1 2 (x!, y!) (8.1)
f(x, y) = rx (1! !x! y)(8.7)
g(x, y) = ry (1! x! "y)(8.8)
(8.1) (8.3) (8.9) v(t) = x(t)! x!, w(t) = y(t)! y!
v(t), w(t) v(t), w(t) (8.9)
(8.3) dv(t)
dw(t)
dt = g(x(t), y(t)) = g(v(t) + x!, w(t) + y!)
2f(x, y), g(x, y) (x!, y!) (v, w) % (0, 0)
f(v + x!, w + y!) = f(x!, y!) + $f
$x (x!, y!)v +
g(v + x!, w + y!) = g(x!, y!) + $g
$x (x!, y!)v +
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dv(t)
dt =
$f
(8.7), (8.8) A (8.12)
A =
%
& =
#
(8.11) (x!, y!) (8.1) A (x!, y!) 8.4. Lotka-Volterra (7.4) (x!, y!) = (0, 0) ,
! 1
! , 0
8.3 Lotka-Volterra (8.1)
(7.8)2 (8.11)A 2 (8.13) v(t) = e"1tp1 + e"2tp2
%1,%2 A p1 A %1p2 A %2 8.5. (8.12) A
(1) A (2) ! > 1, " > 1 A
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(!rx!!! %) (!ry!" ! %)! r2x!y! = 0
(8.14) %2 + r (!x! + "y!)%+ r2x!y! (!" ! 1) = 0
(8.14)
%1, %2 %1 < 0, %2 < 0(8.13) lim t#$
v(t) = 0
x(t) = x!, lim t#$
8.1. Lotka-Volterra (8.1)! > 1, " > 12 (x!, y!) 2 8.4
2 (x, y) = (0, 0) 8.1 2
(x, y) (0, 0) (8.1) 2
x" = rx, y" = ry
7.1 8.2. CoursePower (x!, y!) = (0, 0) (8.1) A
(1) (x!, y!) = (0, 0) (8.1) A = (1) (a) #
$ r 0
0 r
(2) A % = (2) (a) r (b) !r
(3) 7.1 (x!, y!) = (0, 0) (3) (a)
(b) ! < 1, " < 12
!," ! < 1, " < 1 Lotka-Volterra (8.1) 8.2! = " = 0.8 < 1
x(t), y(t)xy
(x(0), y(0)) = (0.19, 0.2)
x(0)Lotka-Volterra
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dx(t)
dy(t)
! > 0, " > 0r > 0
dx(t)
0 = rx! (1! !x! ! y!) ,(8.2a)
0 = ry! (1! x! ! "y!) .(8.2b)
(8.2) Lotka-Volterra (8.1) (0, 0) - 2! 1
! , 0
8.1. !" = 12 x, y (8.1)
f(x, y) =
(8.3)
%
&
f(x, y), g(x, y)x(t), y(t) 8.2. xy 1f(x, y), g(x, y) f(x, y) g(x, y) f(x, y) = 0, g(x, y) = 0 (x, y)
f(x, y) = 0 "# x = 0 1! !x! y = 0,
g(x, y) = 0 "# y = 0 1! x! "y = 0.
#1 : 1! !x! y = 0(8.4)
#2 : 1! x! "y = 0(8.5)
#1, #2 xyf(x, y) = 0, g(x, y) = 0 (x, y) 8.1! > 1, " > 1 #1, #2 8.3. ! < 1, " < 1 #1, #2 xy ! < 1 < "
#1, #2 xy ! > 1, " > 1 8.1xy
1 #1, #2 4
D1 = {(x, y) : 1! !x! y > 0, 1! x! "y > 0}
D2 = {(x, y) : 1! !x! y < 0, 1! x! "y > 0}
D3 = {(x, y) : 1! !x! y < 0, 1! x! "y < 0}
D4 = {(x, y) : 1! !x! y > 0, 1! x! "y < 0} .
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8.1. xy (8.1) #1, #2 Dj (j = 1, 2, 3, 4) . x, y
Dj , j = 1, 2, 3, 4f(x, y), g(x, y) f(x, y) > 0, g(x, y) > 0, (x, y) $ D1
f(x, y) > 0, g(x, y) < 0, (x, y) $ D2
f(x, y) < 0, g(x, y) < 0, (x, y) $ D3
f(x, y) < 0, g(x, y) > 0, (x, y) $ D4
(8.3)xyDj (j = 1, 2, 3, 4) x, y 8.1 8.1x, y 2 (8.6) (x!, y!) =
! " ! 1
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8.1. Lotka-Volterra (8.1)! > 1, " > 12 (x!, y!) 2 8.1. ! < 1, " < 1
8.2
(8.1) 8.1 2 (x!, y!)
8.1 2 (x!, y!) (8.1)
f(x, y) = rx (1! !x! y)(8.7)
g(x, y) = ry (1! x! "y)(8.8)
(8.1) (8.3) (8.9) v(t) = x(t)! x!, w(t) = y(t)! y!
v(t), w(t) v(t), w(t) (8.9)
(8.3) dv(t)
dw(t)
dt = g(x(t), y(t)) = g(v(t) + x!, w(t) + y!)
2f(x, y), g(x, y) (x!, y!) (v, w) % (0, 0)
f(v + x!, w + y!) = f(x!, y!) + $f
$x (x!, y!)v +
g(v + x!, w + y!) = g(x!, y!) + $g
$x (x!, y!)v +
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dv(t)
dt =
$f
(8.7), (8.8) A (8.12)
A =
%
& =
#
(8.11) (x!, y!) (8.1) A (x!, y!) 8.4. Lotka-Volterra (7.4) (x!, y!) = (0, 0) ,
! 1
! , 0
8.3 Lotka-Volterra (8.1)
(7.8)2 (8.11)A 2 (8.13) v(t) = e"1tp1 + e"2tp2
%1,%2 A p1 A %1p2 A %2 8.5. (8.12) A
(1) A (2) ! > 1, " > 1 A
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(!rx!!! %) (!ry!" ! %)! r2x!y! = 0
(8.14) %2 + r (!x! + "y!)%+ r2x!y! (!" ! 1) = 0
(8.14)
%1, %2 %1 < 0, %2 < 0(8.13) lim t#$
v(t) = 0
x(t) = x!, lim t#$
8.1. Lotka-Volterra (8.1)! > 1, " > 12 (x!, y!) 2 8.4
2 (x, y) = (0, 0) 8.1 2
(x, y) (0, 0) (8.1) 2
x" = rx, y" = ry
7.1 8.2. CoursePower (x!, y!) = (0, 0) (8.1) A
(1) (x!, y!) = (0, 0) (8.1) A = (1) (a) #
$ r 0
0 r
(2) A % = (2) (a) r (b) !r
(3) 7.1 (x!, y!) = (0, 0) (3) (a)
(b) ! < 1, " < 12
!," ! < 1, " < 1 Lotka-Volterra (8.1) 8.2! = " = 0.8 < 1
x(t), y(t)xy
(x(0), y(0)) = (0.19, 0.2)
x(0)Lotka-Volterra
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