❆♣♣✳❊✿ Pr♦❣r❛♠♠✐♥❣ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s
❍❛♥s P❡tt❡r ▲❛♥❣t❛♥❣❡♥✶,✷
❙✐♠✉❧❛ ❘❡s❡❛r❝❤ ▲❛❜♦r❛t♦r②✶ ❯♥✐✈❡rs✐t② ♦❢ ❖s❧♦✱ ❉❡♣t✳ ♦❢ ■♥❢♦r♠❛t✐❝s✷
❆✉❣ ✷✶✱ ✷✵✶✻
❍♦✇ t♦ s♦❧✈❡ ❛♥② ♦r❞✐♥❛r② s❝❛❧❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥
u′(t) = αu(t)(✶ − R−✶u(t)) u(✵) = U✵
5 10 15 20 25 30 35 40 45 t 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 u Logistic growth: alpha=0.2, R=1, dt=0.1❊①❛♠♣❧❡s ♦♥ s❝❛❧❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✭❖❉❊s✮
❚❡r♠✐♥♦❧♦❣②✿ ❙❝❛❧❛r ❖❉❊✿ ❛ s✐♥❣❧❡ ❖❉❊✱ ♦♥❡ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥ ❱❡❝t♦r ❖❉❊ ♦r s②st❡♠s ♦❢ ❖❉❊s✿ s❡✈❡r❛❧ ❖❉❊s✱ s❡✈❡r❛❧ ✉♥❦♥♦✇♥ ❢✉♥❝t✐♦♥s ❊①❛♠♣❧❡s✿ u′ = αu ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ u′ = αu
- ✶ − u
R
- ❧♦❣✐st✐❝ ❣r♦✇t❤
u′ + b|u|u = g ❢❛❧❧✐♥❣ ❜♦❞② ✐♥ ✢✉✐❞
❲❡ s❤❛❧❧ ✇r✐t❡ ❛♥ ❖❉❊ ✐♥ ❛ ❣❡♥❡r✐❝ ❢♦r♠✿ u′ = f (u, t)
❖✉r ♠❡t❤♦❞s ❛♥❞ s♦❢t✇❛r❡ s❤♦✉❧❞ ❜❡ ❛♣♣❧✐❝❛❜❧❡ t♦ ❛♥② ❖❉❊ ❚❤❡r❡❢♦r❡ ✇❡ ♥❡❡❞ ❛♥ ❛❜str❛❝t ♥♦t❛t✐♦♥ ❢♦r ❛♥ ❛r❜✐tr❛r② ❖❉❊ u′(t) = f (u(t), t) ❚❤❡ t❤r❡❡ ❖❉❊s ♦♥ t❤❡ ❧❛st s❧✐❞❡ ❝♦rr❡s♣♦♥❞ t♦ f (u, t) = αu, ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ f (u, t) = αu
- ✶ − u
R
- ,
❧♦❣✐st✐❝ ❣r♦✇t❤ f (u, t) = −b|u|u + g, ❜♦❞② ✐♥ ✢✉✐❞ ❖✉r t❛s❦✿ ✇r✐t❡ ❢✉♥❝t✐♦♥s ❛♥❞ ❝❧❛ss❡s t❤❛t t❛❦❡ f ❛s ✐♥♣✉t ❛♥❞ ♣r♦❞✉❝❡ u ❛s ♦✉t♣✉t
❲❤❛t ✐s t❤❡ f (u, t)❄
Pr♦❜❧❡♠✿
- ✐✈❡♥ ❛♥ ❖❉❊✱
√uu′ − α(t)u✸/✷(✶ − u R(t)) = ✵, ✇❤❛t ✐s t❤❡ f (u, t)❄ ❙♦❧✉t✐♦♥✿ ❚❤❡ t❛r❣❡t ❢♦r♠ ✐s u′ = f (u, t)✱ s♦ ✇❡ ♥❡❡❞ t♦ ✐s♦❧❛t❡ u′ ♦♥ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡✿ u′ = α(t)u(✶ − u R(t))
- f (u,t)
❙✉❝❤ ❛❜str❛❝t f ❢✉♥❝t✐♦♥s ❛r❡ ✇✐❞❡❧② ✉s❡❞ ✐♥ ♠❛t❤❡♠❛t✐❝s
❲❡ ❝❛♥ ♠❛❦❡ ❣❡♥❡r✐❝ s♦❢t✇❛r❡ ❢♦r✿ ◆✉♠❡r✐❝❛❧ ❞✐✛❡r❡♥t✐❛t✐♦♥✿ f ′(x) ◆✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥✿ b
a f (x)dx
◆✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ ❛❧❣❡❜r❛✐❝ ❡q✉❛t✐♦♥s✿ f (x) = ✵ ❆♣♣❧✐❝❛t✐♦♥s✿
✶d dx xa s✐♥(wx)✿ f (x) = xa s✐♥(wx)
✷ ✶−✶(x✷ t❛♥❤−✶ x − (✶ + x✷)−✶)dx✿
f (x) = x✷ t❛♥❤−✶ x − (✶ + x✷)−✶✱ a = −✶✱ b = ✶
✸ ❙♦❧✈❡ x✹ s✐♥ x = t❛♥ x✿ f (x) = x✹ s✐♥ x − t❛♥ x