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❆♣♣✳❊✿ 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✐♦♥ ✷ ❙②st❡♠s ♦❢ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✭✈❡❝t♦r ❖❉❊✮

❍♦✇ t♦ s♦❧✈❡ ❛♥② ♦r❞✐♥❛r② s❝❛❧❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ Logistic growth: alpha=0.2, R=1, dt=0.1 1.0 0.9 0.8 0.7 u ′ ( t ) = α u ( t )( ✶ − R − ✶ u ( t )) 0.6 u 0.5 u ( ✵ ) = U ✵ 0.4 0.3 0.2 0.1 0 5 10 15 20 25 30 35 40 45 t

❊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 ′ = α u ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ � ✶ − u � u ′ = α u ❧♦❣✐st✐❝ ❣r♦✇t❤ R 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 ) = α u , ❡①♣♦♥❡♥t✐❛❧ ❣r♦✇t❤ � ✶ − u � f ( u , t ) = α u , ❧♦❣✐st✐❝ ❣r♦✇t❤ R 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 u ′ = α ( t ) u ( ✶ − R ( t )) � �� � f ( u , 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 u ′ = α ( t ) u ( ✶ − R ( t )) � �� � f ( u , 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 u ′ = α ( t ) 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 ) � b ◆✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥✿ a f ( x ) dx ◆✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ ❛❧❣❡❜r❛✐❝ ❡q✉❛t✐♦♥s✿ f ( x ) = ✵ ❆♣♣❧✐❝❛t✐♦♥s✿ dx x a s✐♥ ( wx ) ✿ f ( x ) = x a s✐♥ ( wx ) d ✶ ✷ � ✶ − ✶ ( 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

❲❡ ✉s❡ ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ ❛♣♣r♦①✐♠❛t✐♦♥s t♦ ❞❡r✐✈❛t✐✈❡s t♦ t✉r♥ ❛♥ ❖❉❊ ✐♥t♦ ❛ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ u ′ = f ( u , t ) ❆ss✉♠❡ ✇❡ ❤❛✈❡ ❝♦♠♣✉t❡❞ u ❛t ❞✐s❝r❡t❡ t✐♠❡ ♣♦✐♥ts t ✵ , t ✶ , . . . , t k ✳ ❆t t k ✇❡ ❤❛✈❡ t❤❡ ❖❉❊ u ′ ( t k ) = f ( u ( t k ) , t k ) ❆♣♣r♦①✐♠❛t❡ u ′ ( t k ) ❜② ❛ ❢♦r✇❛r❞ ✜♥✐t❡ ❞✐✛❡r❡♥❝❡✱ u ′ ( t k ) ≈ u ( t k + ✶ ) − u ( t k ) ∆ t ■♥s❡rt ✐♥ t❤❡ ❖❉❊ ❛t t = t k ✿ u ( t k + ✶ ) − u ( t k ) = f ( u ( t k ) , t k ) ∆ t ❚❡r♠s ✇✐t❤ u ( t k ) ❛r❡ ❦♥♦✇♥✱ ❛♥❞ t❤✐s ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ✭❞✐✛❡r❡♥❝❡✮ ❡q✉❛t✐♦♥ ❢♦r u ( t k + ✶ )

❚❤❡ ❋♦r✇❛r❞ ❊✉❧❡r ✭♦r ❊✉❧❡r✬s✮ ♠❡t❤♦❞❀ ✐❞❡❛

❚❤❡ ❋♦r✇❛r❞ ❊✉❧❡r ✭♦r ❊✉❧❡r✬s✮ ♠❡t❤♦❞❀ ✐❞❡❛

❚❤❡ ❋♦r✇❛r❞ ❊✉❧❡r ✭♦r ❊✉❧❡r✬s✮ ♠❡t❤♦❞❀ ♠❛t❤❡♠❛t✐❝s ❙♦❧✈✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ u ( t k + ✶ ) u ( t k + ✶ ) = u ( t k ) + ∆ tf ( u ( t k ) , t k ) ❚❤✐s ✐s ❛ ✈❡r② s✐♠♣❧❡ ❢♦r♠✉❧❛ t❤❛t ✇❡ ❝❛♥ ✉s❡ r❡♣❡❛t❡❞❧② ❢♦r u ( t ✶ ) ✱ u ( t ✷ ) ✱ u ( t ✸ ) ❛♥❞ s♦ ❢♦rt❤✳ ❉✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ♥♦t❛t✐♦♥✿ ▲❡t u k ❞❡♥♦t❡ t❤❡ ♥✉♠❡r✐❝❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ t♦ t❤❡ ❡①❛❝t s♦❧✉t✐♦♥ u ( t ) ❛t t = t k ✳ u k + ✶ = u k + ∆ tf ( u k , t k ) ❚❤✐s ✐s ❛♥ ♦r❞✐♥❛r② ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥ ✇❡ ❝❛♥ s♦❧✈❡✦

■❧❧✉str❛t✐♦♥ ♦❢ t❤❡ ❢♦r✇❛r❞ ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ u ( t ) forward t n − 1 t n t n +1

▲❡t✬s ❛♣♣❧② t❤❡ ♠❡t❤♦❞✦ Pr♦❜❧❡♠✿ ❚❤❡ ✇♦r❧❞✬s s✐♠♣❧❡st ❖❉❊ u ′ = u , t ∈ ( ✵ , T ] ❙♦❧✈❡ ❢♦r u ❛t t = t k = k ∆ t ✱ k = ✵ , ✶ , ✷ , . . . , t n ✱ t ✵ = ✵✱ t n = T ❋♦r✇❛r❞ ❊✉❧❡r ♠❡t❤♦❞✿ u k + ✶ = u k + ∆ t f ( u k , t k ) ❙♦❧✉t✐♦♥ ❜② ❤❛♥❞✿ ❲❤❛t ✐s f ❄ f ( u , t ) = u u k + ✶ = u k + ∆ tf ( u k , t k ) = u k + ∆ tu k = ( ✶ + ∆ t ) u k ❋✐rst st❡♣✿ u ✶ = ( ✶ + ∆ t ) u ✵ ❜✉t ✇❤❛t ✐s u ✵ ❄

❆♥ ❖❉❊ ♥❡❡❞s ❛♥ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✿ u ( ✵ ) = U ✵ ◆✉♠❡r✐❝s✿ ❆♥② ❖❉❊ u ′ = f ( u , t ) ♠✉st ❤❛✈❡ ❛♥ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ u ( ✵ ) = U ✵ ✱ ✇✐t❤ ❦♥♦✇♥ U ✵ ✱ ♦t❤❡r✇✐s❡ ✇❡ ❝❛♥♥♦t st❛rt t❤❡ ♠❡t❤♦❞✦ ▼❛t❤❡♠❛t✐❝s✿ ■♥ ♠❛t❤❡♠❛t✐❝s✿ u ( ✵ ) = U ✵ ♠✉st ❜❡ s♣❡❝✐✜❡❞ t♦ ❣❡t ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥✳ ❊①❛♠♣❧❡✿ u ′ = u ❙♦❧✉t✐♦♥✿ u = Ce t ❢♦r ❛♥② ❝♦♥st❛♥t C ✳ ❙❛② u ( ✵ ) = U ✵ ✿ u = U ✵ e t ✳

❲❡ ❝♦♥t✐♥✉❡ s♦❧✉t✐♦♥ ❜② ❤❛♥❞ ❙❛② U ✵ = ✷✿ u ✶ = ( ✶ + ∆ t ) u ✵ = ( ✶ + ∆ t ) U ✵ = ( ✶ + ∆ t ) ✷ u ✷ = ( ✶ + ∆ t ) u ✶ = ( ✶ + ∆ t )( ✶ + ∆ t ) ✷ = ✷ ( ✶ + ∆ t ) ✷ u ✸ = ( ✶ + ∆ t ) u ✷ = ( ✶ + ∆ t ) ✷ ( ✶ + ∆ t ) ✷ = ✷ ( ✶ + ∆ t ) ✸ u ✹ = ( ✶ + ∆ t ) u ✸ = ( ✶ + ∆ t ) ✷ ( ✶ + ∆ t ) ✸ = ✷ ( ✶ + ∆ t ) ✹ u ✺ = ( ✶ + ∆ t ) u ✹ = ( ✶ + ∆ t ) ✷ ( ✶ + ∆ t ) ✹ = ✷ ( ✶ + ∆ t ) ✺ ✳ ✳ = ✳ ✳ ✳ ✳ u k = ✷ ( ✶ + ∆ t ) k ❆❝t✉❛❧❧②✱ ✇❡ ❢♦✉♥❞ ❛ ❢♦r♠✉❧❛ ❢♦r u k ✦ ◆♦ ♥❡❡❞ t♦ ♣r♦❣r❛♠✳✳✳