個体群動態の数理
• 科目ナンバリングコード：2223011A3
• 開設科目名：個体群動態の数理
• 講義コード：4802000
• 開講期・曜日・時限・教室：前期 水曜日 １・２時限 情 報科学講義室（Ｇ３０２）
• 対象学生：3回生
奈良女子大学理学部・化学生物環境学科
環境科学コース 高須夫悟
2 種系のモデル
2 つの生物集団の関係
ー
互いに負の影響を与える関係を 競争関係という ( Competition )
負の影響とは、片方の存在がもう片方の存在に悪影響（増加率を低下させるなど）を 及ぼすこと
資源（餌）や生存場所を巡って競争している状況が相当する 競争関係にある 2 種の集団密度はどのように変化するのか？
2 種系の競争モデル
種1 種
2
ー
実例 1
Bulmer 1994
実例 2
Brown and Rothery 1994
実例 3
Erickson 1971, Case 1999
Park 1954, Case 1999
南カリフォルニアでの蟻 2 種の競争
Argentine ants > Harvester ants
異なる環境下では勝ち負けが異なる
たいてい勝つ 常に勝つ
T. castaneum < > T. confusum
2 種系の競争モデル
それぞれの集団の個体密度を n 1 , n 2 とする
1 種系のロジスティック増殖のように、集団の増加率が競争相手の個体密 度に比例して減少する場合を考える
種 2 が存在することによる 種 1 の増加率の低下
µ 12 > 0：種 2 が種 1 に及ぼす種間競争の程度を表す
種 2 の個体密度の変化も同様に考える dn 1
dt = r 1
✓
1 n 1 K 1
◆ n 1
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dn 1
dt = r 1
✓
1 n 1
K 1 µ 12 n 2
◆ n 1
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Lotka Volterra の競争モデル
Lotka Volterra の競争モデル
α ij ：種 j が種 i に及ぼす種間競争係数
種間競争係数 α は、自身に対する種内競争係数の強さ 1/K が単位 dn 1
dt = r 1
✓
1 n 1
K 1 µ 12 n 2
◆ n 1
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dn 2
dt = r 2
✓
1 ↵ 21 n 1 + n 2 K 2
◆ n 2
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dn 1
dt = r 1
✓
1 n 1 + ↵ 12 n 2 K 1
◆ n 1
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dn 2
dt = r 2
✓
1 µ 21 n 1 n 2 K 2
◆ n 2
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種1 種
2
種間競争
種間競争 種内競争 種内競争
Lotka Volterra モデルの解析
2 変数の非線形微分方程式は一般に解析的に解くのが困難
グラフを用いて視覚的にモデルの振る舞いを知ることが出来る：
相平面解析（phase plane analysis）もしくはアイソクライン法 Isocline method どのような関数であっても適用が可能
下準備：
横軸に n 1 、縦軸に n 2 の平面（相平面）をとり、時間微分がゼロとなる線を引く。
時間微分がゼロとなる線をヌルクライン null-clineという。
を満たすものが n 1 のヌルクライン
を満たすものが n 2 のヌルクライン
dn 2
dt = r 2
✓
1 ↵ 21 n 1 + n 2 K 2
◆ n 2
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dn 1
dt = r 1
✓
1 n 1 + ↵ 12 n 2 K 1
◆ n 1
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dn 2
dt = 0
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dn 1
dt = 0
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相平面解析
n 1 のヌルクラインは
n 2 のヌルクラインは
n 1 n 2
K 2 K 1 / α 12
dn 1
dt = r 1
✓
1 n 1
K 1 µ 12 n 2
◆
n 1 = 0
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dn 2
dt = r 2
✓
1 ↵ 21 n 1 + n 2 K 2
◆
n 2 = 0
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n 1 = 0, n 1 + ↵ 12 n 2
K 1 = 1
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n 2 = 0, ↵ 21 n 1 + n 2
K 2 = 1
<latexit sha1_base64="aVkd8eyQMnAQ7OiUVtRmYXU8cdg=">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</latexit>
K 1 / α 12 > K 2 , K 2 / α 21 > K 1
平衡点
n 1 n 2
K 1 K 2
K 2 / α 21 K 1 / α 12
0
時間的に変化しない点を平衡点という。
かつ
平衡点は n 1 と n 2 のヌルクラインが交わる点
dn 2
dt = 0
<latexit sha1_base64="S4hUnDUjlgABzbrq6OQKlyn2nK0=">AAACeHichVHLSsNAFD2N7/qqdiO4CRZfm3JbBUVQBDcufVUFlZKkUw2mSUimhRr6A/6ACxeiICp+hht/wEU/QVwqCOLC2zQgKuodZubMmXvunJnRXcv0JVE9prS0trV3dHbFu3t6+/oTA4ObvlP2DJEzHMvxtnXNF5Zpi5w0pSW2XU9oJd0SW/rhUmN/qyI833TsDVl1xV5J27fNomlokql8Irlb9DQjKNj5bC0oyJo6r1I+kaI0haH+BJkIpBDFipO4wi4KcGCgjBIEbEjGFjT43HaQAcFlbg8Bcx4jM9wXqCHO2jJnCc7QmD3kcZ9XOxFr87pR0w/VBp9icfdYqWKUHuiGnumebumR3n+tFYQ1Gl6qPOtNrXDz/cdD66//qko8Sxx8qv70LFHEbOjVZO9uyDRuYTT1laOT5/W5tdFgjC7oif2fU53u+AZ25cW4XBVrp4jzB2S+P/dPsJlNZ6bS2dXp1OJC9BWdGMYIJvi9Z7CIZawgx+dWcYZr3MTeFFUZVyabqUos0iTxJZTsBxPzkTU=</latexit>
dn 1
dt = 0
<latexit sha1_base64="gWra7KFi6fYNCjYu85oUzuVnmJM=">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</latexit>
相平面上の解軌道の向き
n 1 n 2
K 1 K 2
K 2 / α 21 K 1 / α 12
0
(1) (3)
(2) (4)
領域 (1) では、n 1 > 0, n 2 > 0,
n 1 は増加
n 2 は増加
n 1 = 0, n 1 + ↵ 12 n 2
K 1 = 1
<latexit sha1_base64="MJv6XbYy8Y0dvQCUDrleF2oOUs4=">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</latexit>
n 2 = 0, ↵ 21 n 1 + n 2
K 2 = 1
<latexit sha1_base64="aVkd8eyQMnAQ7OiUVtRmYXU8cdg=">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</latexit>
1 ↵ 21 n 1 + n 2
K 2 > 0
<latexit sha1_base64="9EE8vIaUE0j6DCXWggF/PU1DFps=">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</latexit>
1 n 1 + ↵ 12 n 2
K 1 > 0
<latexit sha1_base64="EOfYhSHQG+CWa/VbwY0uOCcnHog=">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</latexit>
dn 1
dt = r 1
✓
1 n 1 + ↵ 12 n 2 K 1
◆
n 1 > 0
<latexit sha1_base64="l+lz56BP1NP3zX8+6feIHmTMo04=">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</latexit>
dn 2
dt = r 2
✓
1 ↵ 21 n 1 + n 2 K 2
◆
n 2 > 0
<latexit sha1_base64="iNN2ixFBJUT6w0JMbv9hgMVJFSA=">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</latexit>
相平面上の解軌道の向き
n 1 n 2
K 1 K 2
K 2 / α 21 K 1 / α 12
0
(1) (3)
(2) (4)
領域 (2) では、n 1 > 0, n 2 > 0,
n 1 は減少
n 2 は増加
n 1 = 0, n 1 + ↵ 12 n 2
K 1 = 1
<latexit sha1_base64="MJv6XbYy8Y0dvQCUDrleF2oOUs4=">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</latexit>
n 2 = 0, ↵ 21 n 1 + n 2
K 2 = 1
<latexit sha1_base64="aVkd8eyQMnAQ7OiUVtRmYXU8cdg=">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</latexit>
1 ↵ 21 n 1 + n 2
K 2 > 0
<latexit sha1_base64="9EE8vIaUE0j6DCXWggF/PU1DFps=">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</latexit>
dn 2
dt = r 2
✓
1 ↵ 21 n 1 + n 2 K 2
◆
n 2 > 0
<latexit sha1_base64="iNN2ixFBJUT6w0JMbv9hgMVJFSA=">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</latexit>
1 n 1 + ↵ 12 n 2
K 1 < 0
<latexit sha1_base64="uQOPUCQj/EDSXqaT4YYu8gFV3A4=">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</latexit>
dn 1
dt = r 1
✓
1 n 1 + ↵ 12 n 2
K 1
◆
n 1 < 0
<latexit sha1_base64="3k9qWwOFz8dRkYYOkLXX9tEtesg=">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</latexit>
相平面上の解軌道の向き
n 1 n 2
K 1 K 2
K 2 / α 21 K 1 / α 12
0
(1) (3)
(2) (4)
領域 (3) では、n 1 > 0, n 2 > 0,
n 1 は増加
n 2 は減少
n 1 = 0, n 1 + ↵ 12 n 2
K 1 = 1
<latexit sha1_base64="MJv6XbYy8Y0dvQCUDrleF2oOUs4=">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</latexit>
n 2 = 0, ↵ 21 n 1 + n 2
K 2 = 1
<latexit sha1_base64="aVkd8eyQMnAQ7OiUVtRmYXU8cdg=">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</latexit>
1 n 1 + ↵ 12 n 2
K 1 > 0
<latexit sha1_base64="EOfYhSHQG+CWa/VbwY0uOCcnHog=">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</latexit>
1 ↵ 21 n 1 + n 2
K 2 < 0
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dn 1
dt = r 1
✓
1 n 1 + ↵ 12 n 2 K 1
◆
n 1 > 0
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dn 2
dt = r 2
✓
1 ↵ 21 n 1 + n 2 K 2
◆
n 2 < 0
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相平面上の解軌道の向き
n 1 n 2
K 1 K 2
K 2 / α 21 K 1 / α 12
0
(1) (3)
(2) (4)
領域 (4) では、n 1 > 0, n 2 > 0,
n 1 は減少
n 2 は減少
n 1 = 0, n 1 + ↵ 12 n 2
K 1 = 1
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n 2 = 0, ↵ 21 n 1 + n 2
K 2 = 1
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1 ↵ 21 n 1 + n 2
K 2 < 0
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dn 2
dt = r 2
✓
1 ↵ 21 n 1 + n 2 K 2
◆
n 2 < 0
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1 n 1 + ↵ 12 n 2
K 1 < 0
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dn 1
dt = r 1
✓
1 n 1 + ↵ 12 n 2
K 1
◆
n 1 < 0
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0.0 0.5 1.0 1.5 2.0 0.0
0.5 1.0 1.5 2.0
数値計算例
矢印：相平面上の各点における解軌道の速度 (dn 1 /dt, dn 2 /dt) のベクトル表示
初期状態が第一象限内にあれば1つの 平衡点（内部平衡点）に収束
解軌道は n
1 のヌルクラインと必ず垂直に交わ
る（ヌルクライン上で n1 の時間微分はゼロ）
解軌道は n
2 のヌルクラインと必ず水平に交わ
る（ヌルクライン上で n2 の時間微分はゼロ）
↵ 21 n 1 + n 2 K 2
= 1
<latexit sha1_base64="1nMVt/GYUABUnkYxlFVPUWwv4U0=">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</latexit>
n 1 + ↵ 12 n 2 K 1
= 1
<latexit sha1_base64="Z+7OcWIAqEPu/5F2u0oUJv9uY6A=">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</latexit>
(n ⇤ 1 , n ⇤ 2 ) =
<latexit sha1_base64="41p7zt8v0FS3Nr5BV2hY23MNoek=">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</latexit>
✓ K 1 ↵ 12 K 2
1 ↵ 12 ↵ 21 , K 2 ↵ 21 K 1 1 ↵ 12 ↵ 21
◆
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K 1 / α 12 > K 2 , K 2 / α 21 > K 1
数値計算例 2
青：n 1 (t) 黄：n 2 (t)
相平面上の解軌道 時刻の関数としての個体密度
両者共存
K 1 / α 12 > K 2 , K 2 / α 21 > K 1
0.0 0.5 1.0 1.5 2.0
0.0 0.5 1.0 1.5 2.0
10 20 30 40 50 t
0.2 0.4 0.6 0.8 1.0 n1, n2
10 20 30 40 50 t
0.2 0.4 0.6 0.8 1.0 n1, n2
10 20 30 40 50 t
0.5 0.6 0.7 0.8 0.9 1.0 n1, n2
0 10 20 30 40 50 t
0.2 0.4 0.6 0.8 1.0 n1, n2
ヌルクラインの 4 通りの交わり型
n 1 n 2
K 1 K 2
K 2 / α 21
K 1 / α 12
0 n 1
n 2
K 1 K 2
K 2 / α 21
K 1 / α 12
0
n 1 n 2
K K 2
K / α K 1 / α 12
0 n 1
n 2
K K 2
K / α K 1 / α 12
0
K 1 / α 12 > K 2 , K 2 / α 21 > K 1 K 1 / α 12 > K 2 , K 2 / α 21 < K 1
K 1 / α 12 < K 2 , K 2 / α 21 > K 1 K 1 / α 12 < K 2 , K 2 / α 21 < K 1
数値計算例 3
相平面上の解軌道 時刻の関数としての個体密度
種 1 のみ生存、種 2 は絶滅
K 1 / α 12 > K 2 , K 2 / α 21 < K 1
0.0 0.5 1.0 1.5
0.0 0.5 1.0 1.5
0 10 20 30 40 50 t
0.2 0.4 0.6 0.8 1.0
n1, n2
10 20 30 40 50 t
0.2 0.4 0.6 0.8 1.0
n1, n2
10 20 30 40 50 t
0.2 0.4 0.6 0.8 1.0
n1, n2
10 20 30 40 50 t
0.2 0.4 0.6 0.8 1.0
n1, n2
青：n 1 (t)
黄：n 2 (t)
数値計算例 4
相平面上の解軌道 時刻の関数としての個体密度
種 1 は絶滅、種 2 のみ生存
K 1 / α 12 < K 2 , K 2 / α 21 > K 1
0 10 20 30 40 50 t
0.2 0.4 0.6 0.8 1.0
n1, n2
10 20 30 40 50 t
0.2 0.4 0.6 0.8 1.0
n1, n2
10 20 30 40 50 t
0.2 0.4 0.6 0.8 1.0
n1, n2
10 20 30 40 50 t
0.2 0.4 0.6 0.8 1.0
n1, n2
0.0 0.5 1.0 1.5
0.0 0.5 1.0 1.5
青：n 1 (t)
黄：n 2 (t)
数値計算例 5
相平面上の解軌道 時刻の関数としての個体密度
初期値に依存してどちらかが絶滅
K 1 / α 12 < K 2 , K 2 / α 21 < K 1
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0 10 20 30 40 50 t
0.2 0.4 0.6 0.8 1.0
n1, n2
10 20 30 40 50 t
0.2 0.4 0.6 0.8 1.0
n1, n2
10 20 30 40 50 t
0.2 0.4 0.6 0.8 1.0
n1, n2
10 20 30 40 50 t
0.2 0.4 0.6 0.8 1.0
n1, n2
青：n 1 (t)
黄：n 2 (t)
Lotka Volterra 競争モデルのまとめ
環境収容量 K 1 , K 2 、種間競争係数 α 12 , α 21 に依存して次の 4 通りが可能 内的自然増加率 r 1 , r 2 は無関係
K 1 / α 12 > K 2 K 1 / α 12 < K 2 K 2 / α 21 > K 1
K 2 / α 21 < K 1
2 種共存 種 1 は絶滅
種 2 のみ 種 2 は絶滅
種 1 のみ
初期条件に依存して どちらかが絶滅
種 2 が種 1 に及ぼす種間競争係数
