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Mục tiêu của luận án Các mô hình liên tục và rời rạc cho hệ sinh thái có yếu tố cạnh tranh là phát triển các mô hình để nghiên cứu các ảnh hưởng của tính phức tạp của môi trường, hành vi của cá thể và cấu trúc tuổi của hệ sinh thái hai loài cạnh tranh
❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❇⑩❈❍ ❑❍❖❆ ❍⑨ ◆❐■ ✖✖✖✖✖✖✖✖✖✖✲ ◆●❯❨➍◆ P❍×❒◆● ❚❍Ị❨ ❈⑩❈ ▼➷ ❍➐◆❍ ▲■➊◆ ❚Ö❈ ❱⑨ ❘❮■ ❘❸❈ ❈❍❖ ❍➏ ❙■◆❍ ❚❍⑩■ ❈➶ ❨➌❯ ❚➮ ❈❸◆❍ ❚❘❆◆❍ ◆❣➔♥❤✿ ❚♦→♥ ❤å❝ ▼➣ sè✿ ✾✹✻✵✶✵✶ ❚➶▼ ❚➁❚ ▲❯❾◆ ⑩◆ ❚■➌◆ ❙➒ ❚❖⑩◆ ❍➴❈ ❍➔ ◆ë✐ ✲ ✷✵✶✽ ❈ỉ♥❣ tr➻♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐✿ ❚r÷í♥❣ ✣↕✐ ữớ ữợ ✶✳ ❚❙✳ ◆❣✉②➵♥ ◆❣å❝ ❉♦❛♥❤ ✷✳ P●❙✳ ❚❙❑❍✳ P❤❛♥ ❚❤à ❍➔ ❉÷ì♥❣ P❤↔♥ ❜✐➺♥ ✶✿ P❤↔♥ ❜✐➺♥ ✷✿ P❤↔♥ ❜✐➺♥ ữủ trữợ ỗ ❣✐→ ❧✉➟♥ →♥ t✐➳♥ s➽ ❝➜♣ ❚r÷í♥❣ t↕✐ ❚r÷í♥❣ ✣↕✐ ỗ t õ t❤➸ t➻♠ ❤✐➸✉ ❧✉➟♥ →♥ t↕✐ t❤÷ ✈✐➺♥✿ ✶✳ ❚❤÷ ✈✐➺♥ ❚↕ ◗✉❛♥❣ ❇û✉ ✕ ❚r÷í♥❣ ✣❍❇❑ ❍➔ ◆ë✐ ✷✳ ❚❤÷ ✈✐➺♥ ◗✉è❝ ❣✐❛ ❱✐➺t ◆❛♠ ▼Ð ✣❺❯ ✶✳ ❚ê♥❣ q ữợ ự ỵ t➔✐ ❙ü t➠♥❣ tr÷ð♥❣ ✈➔ s✉② t❤♦→✐ ❝õ❛ ❝→❝ q✉➛♥ t❤➸ tr♦♥❣ tü ♥❤✐➯♥ ✈➔ sü ✤➜✉ tr❛♥❤ ❝õ❛ ❧♦➔✐ ♥➔② t❤è♥❣ trà ❧♦➔✐ ❦❤→❝ ✤➣ ❧➔ ♠ët ❝❤õ ✤➲ ♥❣❤✐➯♥ ❝ù✉ t❤ó ✈à ✤÷đ❝ q✉❛♥ t➙♠ tr♦♥❣ ♠ët t❤í✐ ❣✐❛♥ ❞➔✐✳ ❱✐➺❝ →♣ ❞ö♥❣ ❝→❝ ❦❤→✐ ♥✐➺♠ t♦→♥ ❤å❝ ✤➸ ❣✐↔✐ t❤➼❝❤ ❝→❝ ❤✐➺♥ t÷đ♥❣ ♥➔② ✤➣ ✤÷đ❝ ❣❤✐ tứ t trữợ t ✤➛✉ t✐➯♥ ✤➦t ♥➲♥ t↔♥❣ ❝❤♦ ✈✐➺❝ ①➙② ❞ü♥❣ ♠æ ❤➻♥❤ ❞ü❛ tr➯♥ t♦→♥ ❤å❝ ❧➔ ▼❛❧t❤✉s ✭✶✼✾✽✮✱ ❱❡r❤✉❧s❡ ✭✶✽✸✽✮✱ P❡❛r❧ ✈➔ ❘❡❡❞ ✭✶✾✵✸✮✱ ✤➦❝ ❜✐➺t ❧➔ ▲♦t❦❛ ✈➔ ❱♦❧t❡r❛ ợ ỳ t q q trồ t ữủ ổ ố tr♦♥❣ ♥❤ú♥❣ ♥➠♠ ✶✾✷✵ ✈➔ ✶✾✸✵✳ ▲♦t❦❛ ✈➔ ❱♦❧t❡rr❛✱ ✤ë❝ ❧➟♣ ✈ỵ✐ ♥❤❛✉✱ ✤➣ ♠ỉ ♣❤ä♥❣ sü ❝↕♥❤ tr❛♥❤ ❣✐ú❛ tú ỗ ỳ ữớ t✐➯♥ ♥❣❤✐➯♥ ❝ù✉ ❤✐➺♥ t÷đ♥❣ t÷ì♥❣ t→❝ ❣✐ú❛ ❝→❝ ❧♦➔✐ ❜➡♥❣ ❝→❝❤ ①➙② ❞ü♥❣ ♠ỉ ❤➻♥❤ ✤ì♥ ❣✐↔♥ ❞ü❛ tr➯♥ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳ ◆❤ú♥❣ ❦➳t q✉↔ ❝õ❛ ♥❣❤✐➯♥ ❝ù✉ õ ỵ ♥❤✐➯♥✱ ❝â r➜t ♥❤✐➲✉ ❤➺ s✐♥❤ t❤→✐ ❝â ②➳✉ tè ❝↕♥❤ tr❛♥❤ ❦❤→❝ ♠➔ ♠æ ❤➻♥❤ ❝↕♥❤ tr❛♥❤ ❝ê ✤✐➸♥ ttrr ổ t t ữủ ỵ ❧➔ ❝â r➜t ♥❤✐➲✉ ❣✐↔ t❤✐➳t tr♦♥❣ ♠æ ❤➻♥❤ ❝õ❛ ▲♦t❦❛✲❱♦❧t❡rr❛✱ ❝❤➥♥❣ ❤↕♥ ♥❤÷ ❣✐↔ ✤à♥❤ r➡♥❣ ♠ỉ✐ tr÷í♥❣ ❧➔ ỗ t tứ ❧➔ ❣✐è♥❣ ♥❤❛✉ ✈➔ sü ❝↕♥❤ tr❛♥❤ ❝❤➾ ✤÷đ❝ t❤➸ ❤✐➺♥ t❤æ♥❣ q✉❛ ❤➺ sè ❝↕♥❤ tr❛♥❤✳ ❚r♦♥❣ ❦❤✐ ✤â✱ ♥❤ú♥❣ ②➳✉ tè ♥❤÷ ♠ỉ✐ tr÷í♥❣✱ ❤➔♥❤ ✈✐ ❝õ❛ ❝→ t❤➸ ❝ơ♥❣ ♥❤÷ ❝→❝ ✤➦❝ t➼♥❤ ❝õ❛ ❝↕♥❤ tr❛♥❤ t❤÷í♥❣ ①✉②➯♥ ①✉➜t ❤✐➺♥ ✈➔ ✤â♥❣ ♠ët ✈❛✐ trá r➜t q✉❛♥ trå♥❣ tr♦♥❣ ❤➺ s✐♥❤ t❤→✐✳ ❱➼ ❞ư ♥❤÷ ❤➔♥❤ ✈✐ ❞✐ ❝÷ ❝õ❛ ♠é✐ ❧♦➔✐ ❧➔ ♠ët ②➳✉ tè r➜t q✉❛♥ trå♥❣ ❝❤♦ sü sè♥❣ ❝á♥ ❝õ❛ ❧♦➔✐ ✤â✳ ❈→❝ ❝→ t❤➸ ❝ò♥❣ ❧♦➔✐ ❤♦➦❝ ❝→❝ ❧♦➔✐ ❦❤→❝ ♥❤❛✉ ❝â t❤➸ ❝â ❝→❝ ✤➦❝ t➼♥❤ ✈➲ ❤➔♥❤ ✈✐ ❦❤→❝ ♥❤❛✉✳ ❍➔♥❤ ✈✐ ❤✉♥❣ ❤➠♥❣ ❝ơ♥❣ ✤÷đ❝ ❝→❝ ❝→ t❤➸ ❤♦❛♥❣ ❞➣ sû ❞ö♥❣ ✤➸ ❝↕♥❤ tr❛♥❤ t❤ù❝ ➠♥✱ ❝❤é ð ❤❛② ❜↕♥ t➻♥❤✳✳✳ ◆❣♦➔✐ r❛✱ ❝→❝ ❝→ t❤➸ ❝ô♥❣ ❝â t❤➸ t❤÷í♥❣ ①✉②➯♥ t❤❛② ✤ê✐ ❤➔♥❤ ✈✐ ❝õ❛ ❝❤ó♥❣ ✤➸ t ự ợ sỹ t ổ trữớ ✤â✱ sü ♣❤→t tr✐➸♥ ❝õ❛ ❝→❝ ♠ỉ ❤➻♥❤ ♠ỵ✐ ❝â t➼♥❤ ✤➳♥ sü ♣❤ù❝ t↕♣ ❝õ❛ ♠ỉ✐ tr÷í♥❣ ✈➔ ❝→❝ ❤➔♥❤ ✈✐ ❝õ❛ ❝→ t❤➸ ✤➣ t❤✉ ❤ót ✤÷đ❝ sü q✉❛♥ t➙♠ ❝õ❛ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝✳ ❙❛✉ ✤➙② ❧➔ ♠ët sè ❝→❝❤ t✐➳♣ ❝➟♥ ❣➛♥ ✤➙②✳ ❈→❝ ♠æ ❤➻♥❤ ✤÷đ❝ ✤÷❛ t❤➯♠ ❝→❝ ②➳✉ tè s❛✉✿ ✲ ❚➼♥❤ ♣❤ù❝ t↕♣ ❝õ❛ ♠ỉ✐ tr÷í♥❣ ✈➔ q✉→ tr➻♥❤ ❞✐ ❝÷ ❝õ❛ ❝→❝ ❝→ t❤➸ tr♦♥❣ ❝→❝ ❤➺ s✐♥❤ t❤→✐ ❝↕♥❤ tr❛♥❤✿ tr♦♥❣ ✤â✱ q✉→ tr➻♥❤ ❝↕♥❤ tr❛♥❤ ✈➔ q✉→ tr➻♥❤ ❞✐ ❝÷ ❝â t❤➸ ❝ị♥❣ t❤❛♥❣ t❤í✐ ❣✐❛♥ ❤♦➦❝ ❦❤→❝ t❤❛♥❣ t❤í✐ ❣✐❛♥✳ ✲ ❚➟♣ t➼♥❤ ❤✉♥❣ ❤➠♥❣ ❝õ❛ ❝→❝ ❝→ t❤➸ tr♦♥❣ ❤➺ s✐♥❤ t❤→✐ ❝↕♥❤ tr❛♥❤✳ ✲ ❈➜✉ tró❝ t✉ê✐ tr♦♥❣ ❤➺ s✐♥❤ t❤→✐ ❝↕♥❤ tr❛♥❤✿ ♥❤â♠ ❝→❝ ❝→ t❤➸ tr÷ð♥❣ t❤➔♥❤ ✈➔ ♥❤â♠ ❝→❝ ❝→ t❤➸ ❝❤÷❛ tr÷ð♥❣ t❤➔♥❤✳ ✷✳ ▼ư❝ ✤➼❝❤✱ ✤è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ▼ö❝ t✐➯✉ ❝õ❛ ❧✉➟♥ →♥ ♥➔② ❧➔ ♣❤→t tr✐➸♥ ❝→❝ ♠ỉ ❤➻♥❤ ✤➸ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ↔♥❤ ❤÷ð♥❣ ❝õ❛ t➼♥❤ ♣❤ù❝ t↕♣ ❝õ❛ ♠ỉ✐ tr÷í♥❣✱ ❤➔♥❤ ✈✐ ❝õ❛ ❝→ t❤➸ ✭❤➔♥❤ ✈✐ ❤✉♥❣ ❤➠♥❣✱ t➟♣ t➼♥❤ s➠♥ ❜➢t✮ ✈➔ ❝➜✉ tró❝ t✉ê✐ ✭♥❤â♠ ❝→❝ ❝→ t❤➸ tr÷ð♥❣ t❤➔♥❤ ✈➔ ♥❤â♠ ❝→❝ ❝→ t❤➸ ❝❤÷❛ tr÷ð♥❣ t❤➔♥❤✮ ❝õ❛ ❤➺ s✐♥❤ t❤→✐ ❤❛✐ ❧♦➔✐ ❝↕♥❤ tr❛♥❤✳ ✣➸ ✤↕t ✤÷đ❝ ♠ư❝ t✐➯✉ ♥➔②✱ ❧✉➟♥ →♥ ✤÷đ❝ ❝❤✐❛ t❤➔♥❤ ✹ ❝ỉ♥❣ ✈✐➺❝ ❝❤➼♥❤ ✶ s❛✉✿ ✲ ❳➙② ❞ü♥❣ ❝→❝ ♠æ ❤➻♥❤ ♣❤➙♥ t➼❝❤ ↔♥❤ ❤÷ð♥❣ t➼♥❤ ♣❤ù❝ t↕♣ ❝õ❛ ♠ỉ✐ tr÷í♥❣ ✈➔ t➟♣ t➼♥❤ ❤✉♥❣ ❤➠♥❣ ❝õ❛ ❝→❝ ❝→ t❤➸ ❝õ❛ ❤➺ ❤❛✐ ❧♦➔✐ ❝↕♥❤ tr❛♥❤✳ ✲ ❳➙② ❞ü♥❣ ❝→❝ ♠æ ❤➻♥❤ ♣❤➙♥ t➼❝❤ ↔♥❤ ❤÷ð♥❣ ❝õ❛ ❝➜✉ tró❝ t✉ê✐ ✭♥❤â♠ ❝→❝ ❝→ t❤➸ tr÷ð♥❣ t❤➔♥❤ ✈➔ ♥❤â♠ ❝→❝ ❝→ t❤➸ ❝❤÷❛ tr÷ð♥❣ t❤➔♥❤✮ ✤è✐ ✈ỵ✐ ❤➺ s✐♥❤ t❤→✐ ❝↕♥❤ tr❛♥❤✳ ✲ ❳➙② ỹ ổ ỹ tr ổ ỗ t❤à ✤➽❛ ✤➸ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❤➺ s✐♥❤ t❤→✐ ❝↕♥❤ tr❛♥❤✳ ✲ ❚❤✐➳t ❦➳✱ ❝➔✐ ✤➦t ✈➔ t✐➳♥ ❤➔♥❤ ❝→❝ ♠ỉ ♣❤ä♥❣✳ ✸✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ❚r♦♥❣ ❧✉➟♥ →♥✱ ❝❤ó♥❣ tỉ✐ sû ❞ư♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ s❛✉✿ ✲ ❈→❝ ♣❤÷ì♥❣ ♣❤→♣ ♠ỉ ❤➻♥❤ ❤â❛ ❞ü❛ tr➯♥ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❞ü❛ tr➯♥ ❝→ t❤➸ ✤÷đ❝ sû ❞ư♥❣ ✤➸ ♠ỉ ♣❤ä♥❣ ❝→❝ ❤➺ s✐♥❤ t❤→✐ ð ❝→❝ ♠ù❝ ✤ë ♣❤ù❝ t↕♣ ✈➔ ❝ư t❤➸ ❦❤→❝ ♥❤❛✉✳ ✲ ❈→❝ ♣❤÷ì♥❣ ♣❤→♣ ❤➺ ✤ë♥❣ ❧ü❝ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ✤÷đ❝ sû ❞ư♥❣ ✤➸ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ♠ỉ ❤➻♥❤ t♦→♥ ❤å❝ t❤✉ ✤÷đ❝✳ ✣➦❝ ❜✐➺t✱ ♣❤÷ì♥❣ ♣❤→♣ tê ❤đ♣ ❜✐➳♥ ✤÷đ❝ ❧ü❛ ❝❤å♥ ✤➸ ❧➔♠ ❣✐↔♠ sü ♣❤ù❝ t↕♣ ❝õ❛ ♠æ ❤➻♥❤✳ ữỡ q ỵ tt ỗ t❤à ✤÷đ❝ ①❡♠ ①➨t ✤➸ ♥❣❤✐➯♥ ❝ù✉ ♠ët sè ♠ỉ ỗ t ữủ t r tứ ổ ❞ü❛ tr➯♥ ❝→ t❤➸✳ ✹✳ Þ ♥❣❤➽❛ ✈➔ ❦➳t q✉↔ ❝õ❛ ❧✉➟♥ →♥ ▲✉➟♥ →♥ tr➻♥❤ ❜➔② ❝→❝ ♠æ ❤➻♥❤ ✈➔ ♠ỉ ♣❤ä♥❣ ❦❤→❝ ♥❤❛✉ ❝â t❤➸ ✤÷đ❝ →♣ ❞ư♥❣ tr ự ỵ tt ụ ữ ự tỹ ♥❣❤✐➺♠ tr♦♥❣ ❝→❝ ❤➺ s✐♥❤ t❤→✐ ❝↕♥❤ tr❛♥❤✳ ❱➲ ♠➦t ỵ tt t tr t ổ ởt sè ♠ỉ ❤➻♥❤ ❧✐➯♥ tư❝ ✈➔ rí✐ r↕❝ ♠ỉ t↔ ❤➺✳ ❱➲ ♠ỉ ❤➻♥❤ ❧✐➯♥ tư❝✿ ♠ỉ t↔ ❤❛✐ ❧♦➔✐ ❝↕♥❤ tr❛♥❤ ♥❤❛✉✱ ❝ò♥❣ ❦❤❛✐ t❤→❝ ♠ët t➔✐ ♥❣✉②➯♥ ❝❤✉♥❣ ợ ữủ tr ổ ❤➻♥❤ rí✐ r↕❝✿ ♠ỉ t↔ ❤➺ s✐♥❤ t❤→✐ ❝↕♥❤ tr❛♥❤ ỳ tú ỗ ổ ọ ❝→❝ ♠ỉ ❤➻♥❤ ✭❧✐➯♥ tư❝ ✈➔ rí✐ r↕❝✮ ✤÷đ❝ t❤✐➳t ❦➳ ✤➸ ♠✐♥❤ ❤å❛ ✈➔ t❤û ♥❣❤✐➺♠ ❝→❝ ❦à❝❤ ❜↔♥ ❦❤→❝ ♥❤❛✉✳ ❱➲ ♠➦t ù♥❣ ❞ö♥❣✱ ❧✉➟♥ →♥ ✤➣ tr➻♥❤ ❜➔② ♠ët sè ♠æ ❤➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ❤❛✐ ❤➺ s✐♥❤ t❤→✐ ❝ö t❤➸✿ ✭✶✮ ❤➺ s✐♥❤ t❤→✐ ❝↕♥❤ tr❛♥❤ ❣✐ú❛ ❧♦➔✐ ❝→ ❚❤✐♦❢ ✈➔ ❧♦➔✐ ❜↕❝❤ t✉ë❝ ð ✈ò♥❣ ❜✐➸♥ ❙❡♥❡❣❛❧ ✈➔ ✭✷✮ ❤➺ s✐♥❤ t❤→✐ ❝↕♥❤ tr❛♥❤ ❣✐ú❛ ❧ó❛ ✈➔ r➛② ♥➙✉✳ ✺✳ ❈➜✉ tró❝ ❝õ❛ ❧✉➟♥ →♥ ◆❣♦➔✐ ♣❤➛♥ ▼ð ✤➛✉✱ ❑➳t ❧✉➟♥ ✈➔ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ❧✉➟♥ →♥ ✤÷đ❝ ❝❤✐❛ ❧➔♠ ❜è♥ ❝❤÷ì♥❣✿ ✿ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ❝↕♥❤ tr❛♥❤ tr♦♥❣ ❤➺ s✐♥❤ t❤→✐ ữợ t ự s t tr ỗ ổ tư❝ ✈➔ ❝→❝ ♠ỉ ❤➻♥❤ rí✐ r↕❝✳ P❤÷ì♥❣ ♣❤→♣ ❤➔♠ ỵ t ữỡ tờ ❤đ♣ ❜✐➳♥ ❝ơ♥❣ ✤÷đ❝ tr➻♥❤ ❜➔② ❝ư t❤➸ tr♦♥❣ ❝❤÷ì♥❣ ♥➔②✳ ✿ tr➻♥❤ ❜➔② ♠ët sè ♠ỉ ❤➻♥❤ ❧✐➯♥ tư❝ trữớ ủ tr ũ ởt ỗ tự ợ ữủ tr ♥❤❛✉✳ ✿ tr➻♥❤ ❜➔② ♠ët sè ♠ỉ ❤➻♥❤ rí✐ r↕❝ tú ỗ tứ ổ t ổ ỗ t t ữủ tứ ổ ❤➻♥❤ ❝→ t❤➸✳ ❈❤÷ì♥❣ ✶ ❈❤÷ì♥❣ ✷ ❈❤÷ì♥❣ ✸ ✷ ❈❤÷ì♥❣ ✹ ✿ tr➻♥❤ ❜➔② ♠ët sè ♠ỉ ❤➻♥❤ ❝❤♦ ❝→❝ ❤➺ s✐♥❤ t❤→✐ ❝ö t❤➸✿ ✭✶✮ ❤➺ s✐♥❤ t❤→✐ ❝↕♥❤ tr❛♥❤ ❣✐ú❛ ❧♦➔✐ ❝→ ❚❤✐♦❢ ✈➔ ❧♦➔✐ ❜↕❝❤ t✉ë❝ ✈➔ ✭✷✮ ❤➺ s✐♥❤ t❤→✐ ❝↕♥❤ tr❛♥❤ ❣✐ú❛ ❧ó❛ ✈➔ r➛② ♥➙✉✳ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ →♥ ❞ü❛ ✈➔♦ ❜è♥ ❜➔✐ ❜→♦✱ ✤÷đ❝ ❧✐➺t ❦➯ ð ✱ tr♦♥❣ ✤â ❝→❝ ❜➔✐ ❬✶❪✱❬✷❪ ✤÷đ❝ ✤➠♥❣ tr➯♥ t↕♣ ❝❤➼ t❤✉ë❝ ♥❤â♠ ✭❙❈■✮✱ ❜➔✐ ❜→♦ ❬✸❪ ✤➣ ✤÷đ❝ ♥❤➟♥ ✤➠♥❣ ✈➔ ❜➔✐ ❜→♦ ❬✹❪ ✤➣ ❣û✐✳ ♠ư❝ ❝ỉ♥❣ tr➻♥❤ ✤➣ ❝ỉ♥❣ ❜è ❝õ❛ ❧✉➟♥ →♥✧ ✧❉❛♥❤ ❈❤÷ì♥❣ ✶ ❑■➌◆ ❚❍Ù❈ ❚✃◆● ◗❯❆◆ ✶✳✶ ❈↕♥❤ tr❛♥❤ tr♦♥❣ ❤➺ s✐♥❤ t❤→✐ ❈↕♥❤ tr❛♥❤ ✤â♥❣ ♠ët ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ s✐♥❤ t❤→✐ q✉➛♥ t❤➸✳ ◆➳✉ ❝→❝ ✤è✐ t❤õ ❝↕♥❤ tr❛♥❤ ❧➔ ❝ò♥❣ ♠ët ❧♦➔✐ t❤➻ ❝✉ë❝ ❝↕♥❤ tr❛♥❤ ✤÷đ❝ ❣å✐ ❧➔ ❝↕♥❤ tr❛♥❤ ♥ë✐ t↕✐ ✭✐♥tr❛s♣❡❝✐❢✐❝ ❝♦♠♣❡t✐t✐♦♥✮✳ ❈↕♥❤ tr❛♥❤ ❝ò♥❣ ❧♦➔✐ ❝â t❤➸ ❧➔ tr❛♥❤ ❣✐➔♥❤ ♥ì✐ ð✱ ❜↕♥ t➻♥❤✱ ❤♦➦❝ ❝↕♥❤ tr❛♥❤ t❤ù❝ tr ũ tữớ ỗ t➔✐ ♥❣✉②➯♥ t❤ù❝ ➠♥ ❝❤♦ ♠é✐ ❝→ t❤➸✱ ✈➔ ❞♦ ✤â ❧➔♠ ❣✐↔♠ t✛ ❧➺ t➠♥❣ tr÷ð♥❣ ✈➔ ♣❤→t tr✐➸♥ ❝õ❛ ❝→❝ ❝→ t❤➸✳ ◆➳✉ ❝→❝ ✤è✐ t❤õ ❝↕♥❤ tr❛♥❤ ❧➔ ❝õ❛ ❝→❝ ❧♦➔✐ ❦❤→❝ ♥❤❛✉ t❤➻ ✤÷đ❝ ❣å✐ ❧➔ tr trs tt ỵ ỡ ❧➔ sü ①✉➜t ❤✐➺♥ ❝õ❛ ❧♦➔✐ ♥➔② ↔♥❤ ❤÷ð♥❣ t✐➯✉ ❝ü❝ ✤➳♥ sü t➠♥❣ tr÷ð♥❣ ✈➔ ♣❤→t tr✐➸♥ ❝õ❛ ❧♦➔✐ ổ tử ữợ ổ ❤â❛ ❞ü❛ tr➯♥ ♣❤÷ì♥❣ tr➻♥❤ ✭❊q✉❛t✐♦♥✲❇❛s❡❞ ▼♦❞❡❧✐♥❣✲ ❊❇▼✮ ❝â ❧à❝❤ sû ❧➙✉ ✤í✐ tr♦♥❣ ♥❣❤✐➯♥ ❝ù✉ ❤➺ s✐♥❤ t❤→✐✳ ✣➙② ❧➔ ♠ët ❝ỉ♥❣ ❝ư r➜t ❤ú✉ ❤✐➺✉ ❝❤♦ ♣❤➨♣ ♥❣❤✐➯♥ ❝ù✉ ❞→♥❣ ✤✐➺✉ t✐➺♠ ❝➟♥ ❝õ❛ ❤➺ ✈➔ ✤♦ ✤â ❝✉♥❣ ❝➜♣ ❝→❝ t❤æ♥❣ t✐♥ ð ♠ù❝ ✤ë q✉➛♥ t❤➸ ❤❛② q✉➛♥ ①➣✳ ❈→❝ ♣❤÷ì♥❣ tr➻♥❤ t❤÷í♥❣ sû ❞ư♥❣ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✱ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥✱ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❤❛② ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ♥❣➝✉ ♥❤✐➯♥✳ ✶✳✸ ❈→❝ ♠ỉ ❤➻♥❤ rí✐ r↕❝ ❈→❝ ♠æ ❤➻♥❤ ❞ü❛ tr➯♥ ❝→ t❤➸ ▼æ ❤➻♥❤ ❤â❛ ❞ü❛ tr➯♥ ❝→ t❤➸ ✭■♥❞✐✈✐❞✉❛❧✲❇❛s❡❞ ▼♦❞❡❧✐♥❣✲ ■❇▼✮ ❧➔ ♠ët ❧♦↕✐ ♠æ t ởt ổ ỗ t❤➔♥❤ ♣❤➛♥ ❝ì ❜↔♥✿ ♠ỉ✐ tr÷í♥❣ ✈➔ ❝→ t❤➸✳ ❈→❝ ❤➔♥❤ ✈✐ ❝õ❛ ❝→ t❤➸ ❝ơ♥❣ ♥❤÷ t÷ì♥❣ t→❝ ❝õ❛ ú ợ ợ ổ trữớ ữủ ❝ư t❤➸ tr♦♥❣ ♠ët ❝❤÷ì♥❣ tr➻♥❤ ♠→② t➼♥❤✳ ▲♦↕✐ ♠ỉ ❤➻♥❤ ♥➔② t❤÷í♥❣ ✤÷đ❝ sû ❞ư♥❣ ✤➸ ♥❣❤✐➯♥ ❝ù✉ ↔♥❤ ❤÷ð♥❣ ❝õ❛ ❤➔♥❤ ✈✐ ❝õ❛ ❝→❝ ❝→ t❤➸ ð ✤à❛ ♣❤÷ì♥❣ ❧➯♥ ❜ù❝ tr❛♥❤ ✤ë♥❣ ❧ü❝ ❝õ❛ t♦➔♥ ❜ë ❤➺✳ ổ ỹ tr ỗ t ổ õ ỹ tr ỗ t s rs ởt ổ ỗ t tr õ ộ ✤➾♥❤ ❝â ♠ët ❜→♥ ❦➼♥❤ ✈ị♥❣ ↔♥❤ ❤÷ð♥❣ ✈➔ ❤❛✐ ✤➾♥❤ ❝â ❝↕♥❤ ❦❤✐ ❤❛✐ ✈ị♥❣ ↔♥❤ ❤÷ð♥❣ ❣✐❛♦ ♥❤❛✉✳ ▲♦↕✐ ♠æ ❤➻♥❤ ♥➔② ❝❤♦ ♣❤➨♣ t➼♥❤ t♦→♥ ❝→❝ ✤➦❝ t ữỡ t ị tữ sû ❞ư♥❣ ❧♦↕✐ ♠ỉ ❤➻♥❤ tr♦♥❣ ❤➺ s✐♥❤ t❤→✐ ♥➡♠ ð ✈✐➺❝ ❝♦✐ ♠é✐ ❝→ t❤➸ ❧➔ ♠ët ✤➾♥❤ tr♦♥❣ ỗ t t tr ❣✐ú❛ ❝❤ó♥❣ ❝â ❝↕♥❤✳ ✶✳✹ P❤÷ì♥❣ ♣❤→♣ ❤➔♠ ▲②❛♣✉♥♦✈ ✈➔ ỵ t Pữỡ tờ ủ ❜✐➳♥ P❤÷ì♥❣ ♣❤→♣ tê ❤đ♣ ❜✐➳♥ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ♥❤â♠ ❝õ❛ ●✐→♦ s÷ P✐❡rr❡ ❆✉❣❡r ✈➔ ❝ë♥❣ sü ♥➠♠ ✷✵✵✽✳ ❈→❝ ♠æ ❤➻♥❤ ữủ t tở ởt ợ ữỡ tr tữớ ợ t tớ õ t ữợ s dn = f (n) + s(n) dτ ✭✶✳✶✮ ✈ỵ✐ n ∈ Rm ✱ ❝→❝ →♥❤ ①↕ f ✈➔ s ❜✐➸✉ ❞✐➵♥ t÷ì♥❣ ù♥❣ ❝❤♦ ❝→❝ q✉→ tr➻♥❤ ♥❤❛♥❤ ✈➔ ❝❤➟♠✳ ❧➔ t❤❛♠ sè ❞÷ì♥❣ ♥❤ä ❜✐➸✉ ❞✐➵♥ t✛ ❧➺ ❣✐ú❛ ❤❛✐ t❤❛♥❣ t❤í✐ ❣✐❛♥✳ tỹ ữủ ữỡ tờ ủ trữợ t✐➯♥ ❤➺ ✭✶✳✶✮ ❝➛♥ ✤÷❛ ✈➲ ❞↕♥❣ ❜✐➸✉ ❞✐➵♥ ♥❤❛♥❤✲❝❤➟♠ ♥❤÷ s❛✉ ❜➡♥❣ ❝→❝❤ ❜✐➳♥ ✤ê✐ ❜✐➳♥ n ∈ Rm → (x, y) ∈ Rm−k × Rk ✿ dx = F (x, y) + S(x, y), dτ ✭✶✳✷✮ dy = G(x, y), dτ ✈ỵ✐ F, S, G ❧➔ ❝→❝ ❤➔♠ ✤õ trì♥✱ x ❜✐➸✉ ❞✐➵♥ ❝→❝ ❜✐➳♥ tr♦♥❣ ❝→❝ q✉→ tr➻♥❤ ♥❤❛♥❤ ✈➔ y ❜✐➸✉ ❞✐➵♥ ❝→❝ ❜✐➳♥ tr♦♥❣ ❝→❝ q✉→ tr➻♥❤ ❝❤➟♠✳ ❑❤✐ ✤â✱ ữợ ữỡ tờ ủ ữợ = tr ữỡ tr ✤➛✉ t✐➯♥ ❝õ❛ ❤➺ ♥❤❛♥❤✲❝❤➟♠ ✭✶✳✷✮✳ ❚ø ♣❤÷ì♥❣ tr➻♥❤ t❤ù ✷ ❝â t❤➸ ❝♦✐ y ❧➔ ❤➡♥❣ sè✱ t➻♠ ❝→❝ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ê♥ ✤à♥❤ dx t✐➺♠ ❝➟♥ x∗ (y) d = F (x, y) ữợ ❚rð ❧↕✐ ❤➺ ✤ë♥❣ ❧ü❝ tr♦♥❣ t❤❛♥❣ t❤í✐ ❣✐❛♥ ❝❤➟♠✳ ❚❤❛② t❤➳ x∗ (y) ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ t❤ù ❤❛✐ ❝õ❛ ❤➺ ♥❤❛♥❤✲❝❤➟♠ ✭✶✳✷✮✱ t❤✉ ✤÷đ❝ ❤➺ rót ❣å♥✿ dy = G(x∗ (y), y), dt ✈ỵ✐ t = τ ❜✐➸✉ ❞✐➵♥ tớ ữợ tr❛ ❤❛✐ ✤✐➲✉ ❦✐➺♥✿ ✭❍✶✮ ❤➺ ✭✶✳✸✮ ❧➔ ê♥ ✤à♥❤ ❝➜✉ tró❝ ✤à❛ ♣❤÷ì♥❣✴t♦➔♥ ❝ư❝ ①✉♥❣ q✉❛♥❤ ❝→❝ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ✈➔ ✭❍✷✮ ❧➔ ✤õ ♥❤ä✱ ✤↔♠ ❜↔♦ ❝❤♦ sü ①➜♣ ①➾✳ ❈❤÷ì♥❣ ✷ ❈⑩❈ ▼➷ ❍➐◆❍ ▲■➊◆ ❚Ư❈ ❈❍❖ ìẹ ợ t ❤➺ ❝↕♥❤ tr❛♥❤ ✷✳✷ ▼æ ❤➻♥❤ ❝↕♥❤ tr❛♥❤ ❝ê ✤✐➸♥ ▼ỉ ❤➻♥❤ ❝↕♥❤ tr❛♥❤ ❝ê ✤✐➸♥ ✤÷đ❝ ❝❤♦ ♥❤÷ s❛✉✿ dR = [γ(R) − a1 C1 R − a2 C2 R] dt dC1 = [−d1 C1 + a1 e1 RC1 ] dt dC2 = [−d2 C2 + a2 e2 RC2 ] , dt ✭✷✳✶✮ ✈ỵ✐ ❤➔♠ γ(R) ❜✐➸✉ ❞✐➵♥ sỹ t trữ ỗ tự ỗ tự ➠♥ ❝â ❞↕♥❣ s✐♥❤ ❤å❝ ✭❜✐♦t✐❝✮✱ t❛ ❝â γ(R) = rR(1 R/K) ợ r K tữỡ ự tè❝ ✤ë t➠♥❣ tr÷ð♥❣ ✈➔ sù❝ ❝❤ù❛ ♠ỉ✐ tr÷í♥❣ ❝õ❛ ỗ tự (R) = r(S R) ỗ tự ổ s t ợ r tố ỗ t S ỗ t tữỡ tỹ ữ ❝❤ù❛ ♠ỉ✐ tr÷í♥❣✳ di ❧➔ t❤❛♠ sè ❜✐➸✉ ❞✐➵♥ t✛ ❧➺ ❝❤➳t tü ♥❤✐➯♥ ❝õ❛ ❧♦➔✐ i✱ ❜✐➸✉ ❞✐➵♥ t t ỗ i ố ợ ỗ t❤ù❝ ➠♥ ❛♥❞ ei ❧➔ t❤❛♠ sè ❧✐➯♥ q✉❛♥ ✤➳♥ ❝❤✉②➸♥ ❤â❛ t❤ù❝ ➠♥ ❝õ❛ ❧♦➔✐ i✱ i ∈ {1, 2}✳ ✣✐➲✉ ❦✐➺♥ ❝❤♦ ❝↕♥❤ tr❛♥❤ ❜➜t ✤è✐ ①ù♥❣ t↕✐ ✤à❛ ♣❤÷ì♥❣✱ ð ✤➙② C1 ❧➔ ❧♦➔✐ ❝↕♥❤ tr❛♥❤ ♠↕♥❤ ✈➔ C2 ❧➔ ❧♦➔✐ ❝↕♥❤ tr❛♥❤ ②➳✉✱ ✤÷đ❝ ❝❤♦ ❜ð✐ d1 d2 < ♠✐♥ R∗ , a1 e1 a2 e2 , ✭✷✳✷✮ ð ✤➙② R∗ ❧➔ ♠ù❝ ❝➙♥ ❜➡♥❣ ❝õ❛ ỗ tự ổ õ t tú R = K ỗ tự t R = S ỗ tự t ổ ợ ữủ trố tr ú tổ tt q tr ữ r ỡ s ợ q✉→ tr➻♥❤ t➠♥❣ tr÷ð♥❣ ✈➔ q✉→ tr➻♥❤ ❝↕♥❤ tr❛♥❤✳ ▼ỉ ❤➻♥❤ ❝↕♥❤ tr❛♥❤ ❝ê ✤✐➸♥ ✭✷✳✶✮ ✤÷đ❝ ✺ sû ❞ư♥❣ ❧↕✐ tr♦♥❣ ✈ò♥❣ ❝â sü ❝↕♥❤ tr❛♥❤✳ ❚↕✐ ❝→❝ ✈ò♥❣ ❦❤ỉ♥❣ ❝â ❝↕♥❤ tr❛♥❤✱ ❝❤ó♥❣ tỉ✐ ❜ê s✉♥❣ ❝→❝ t❤➔♥❤ ♣❤➛♥ t❤➸ ❤✐➺♥ t✛ ❧➺ ❝❤➳t ❝õ❛ ❧♦➔✐ ✈➔ q✉→ tr➻♥❤ ❞✐ ❝÷ ❣✐ú❛ ❝→❝ ✈ị♥❣✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ ♠ỉ ❤➻♥❤ ✤÷đ❝ ❝❤♦ ❜ð✐ ❤➺✿ dR = ε γ(R) − a1 C1 R − a2 C2C R dτ dC1 dτ = ε[−d1 C1 + a1 e1 RC1 ] ✭✷✳✸✮ dC2C = (kC2N − (αC1 + α0 )C2C ) + ε[−d2C C2C + a2 e2 RC2C ] dτ dC2N = ((αC1 + α0 )C2C − kC2N ) − εd2N C2N , dτ ✈ỵ✐ ❝→❝ t❤❛♠ sè ♠ỵ✐ C2C ✭t÷ì♥❣ ù♥❣ C2N ✮ ✈➔ d2C ✭t÷ì♥❣ ù♥❣ d2N ✮ ❜✐➸✉ ❞✐➵♥ ♠➟t ✤ë ✈➔ t✛ ❧➺ ❝❤➳t tü tr ũ tr tữỡ ự ợ ✈ị♥❣ ❦❤ỉ♥❣ ❝↕♥❤ tr❛♥❤✮✳ k ❧➔ t❤❛♠ sè ❜✐➸✉ ❞✐➵♥ t✛ ❧➺ ❞✐ ❝÷ ❜➻♥❤ q✉➙♥ tø ✈ị♥❣ ❦❤ỉ♥❣ ❝↕♥❤ tr❛♥❤ ✤➳♥ ✈ò♥❣ ❝↕♥❤ tr❛♥❤✳ αC1 + α0 ❜✐➸✉ ❞✐➵♥ ❞✐ ❝÷ ♣❤ư t❤✉ë❝ ♠➟t ✤ë tø ✈ị♥❣ ❝↕♥❤ tr❛♥❤ ✤➳♥ ✈ị♥❣ ❦❤ỉ♥❣ ❝↕♥❤ tr❛♥❤✳ Ð ✤➙② α ❜✐➸✉ ❞✐➵♥ ✤ë ❧ỵ♥ ❝õ❛ sü ♣❤ư t❤✉ë❝ ♠➟t ✤ë ❝õ❛ q✉→ tr➻♥❤ ❞✐ ❝÷✳ ❱➼ ❞ư ♥➳✉ ❝â q✉→ ♥❤✐➲✉ ❝→ t❤➸ ▲❙❊ tr♦♥❣ ✈ò♥❣ ❝↕♥❤ tr❛♥❤ t❤➻ ❝→❝ ❝→ t❤➸ ▲■❊ s➩ ❝â ①✉ t❤➳ rí✐ ❦❤ä✐ ✈ị♥❣ ❝↕♥❤ tr❛♥❤ ✤➸ ❞✐ ❝÷ ✤➳♥ ✈ị♥❣ ❦❤ỉ♥❣ ❝â ❝↕♥❤ tr❛♥❤✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ α = 0✱ sü ❞✐ ❝÷ ❧➔ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ♠➟t ✤ë ✈ỵ✐ tè❝ ✤ë α0 ✳ ❚❤❛♠ sè ε ❜✐➸✉ ❞✐➵♥ t✛ ❧➺ ❣✐ú❛ ❤❛✐ t❤❛♥❣ t❤í✐ ❣✐❛♥ t = ετ ✱ t ❧➔ t❤❛♥❣ t❤í✐ ❣✐❛♥ ❝❤➟♠ ✈➔ τ ❧➔ t❤❛♥❣ t❤í✐ ❣✐❛♥ ♥❤❛♥❤✳ ❈→❝ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❝↕♥❤ tr❛♥❤ ❜➜t ✤è✐ ①ù♥❣ ✭✷✳✷✮ ❧ó❝ ♥➔② trð t❤➔♥❤✿ d1 d2C < ♠✐♥ R∗ , ✭✷✳✹✮ a1 e1 a2 e2 ▼ỉ ❤➻♥❤ rót ❣å♥ ❙û ❞ư♥❣ ữợ ữỡ tờ ủ ❤➺ ✭✷✳✸✮ ✤÷đ❝ ✤÷❛ ✈➲ ❤➺ rót ❣å♥ s❛✉ dR a2 k = γ(R) − a1 RC1 − RC2 dt H(C 1) dC1 ✭✷✳✺✮ = C1 [−d1 + a1 e1 R] dt C2 dC2 = − kd2C + d2N (αC1 + α0 ) + a2 e2 kR dt H(C1 ) ✈ỵ✐ C2 = C2C + C2N ✳ ❙ü ê♥ ✤à♥❤ t♦➔♥ ❝ư❝ ❝õ❛ ♠ỉ ❤➻♥❤ rót ❣å♥ ✭✷✳✺✮ ✻ ❑❤✐ R∗ < R2+ = (kd2C + α0 d2N )/a2 e2 k✱ ❝→❝ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ❧➔ (R∗ , 0, 0)✱ (0, 0, 0) ✭❝❤♦ tr÷í♥❣ ❤đ♣ ỗ tự s (R1+ , C1+ , 0) ✈ỵ✐ R1+ = d1 /a1 e1 ✱ C1+ = γ(R1+ )/a1 R1+ ❞÷ì♥❣ ❦❤✐ R1+ < R∗ ✭tø ✤✐➲✉ ❦✐➺♥ ✭✷✳✹✮✮✳ ❑❤✐ R∗ > R2+ ✱ ❝→❝ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ❧➔ (R∗ , 0, 0)✱ (R2+ , 0, C2+ ) (0, 0, 0) trữớ ủ ỗ tự ➠♥ ❦❤æ♥❣ ❧➔ s✐♥❤ ❤å❝✮ ✈➔ (R1+ , C1+ , 0) ợ C2+ = (R2+ )H(0)/a2 kR2+ ỵ ✷✳✸✳✶✳ (R1+ , C1+ , 0) ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ t♦➔♥ ❝ö❝ tr♦♥❣ R3+ ✳ ❚â♠ ❧↕✐✱ tr♦♥❣ ❜➜t ❦ý tr÷í♥❣ ❤đ♣ ♥➔♦✱ ▲❙❊ ❧✉ỉ♥ ❧➔ ❧♦➔✐ ❝❤✐➳♥ t❤➢♥❣ tr➯♥ t♦➔♥ ❝ư❝✳ ◆â✐ ❝→❝❤ ❦❤→❝✱ ❝❤✐➳♥ ❧÷đ❝ trè♥ tr→♥❤ ❝õ❛ ▲■❊ ❧➔ ❦❤æ♥❣ ❤✐➺✉ q✉↔ ✤➸ t❤♦→t ❦❤ä✐ t✉②➺t ổ ợ ữủ r ♣❤➛♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ①❡♠ ①➨t ❝❤✐➳♥ ❧÷đ❝ t❤ù ❤❛✐ ❧➔ ❝→❝ ❝→ t❤➸ ▲■❊ trð ♥➯♥ r➜t ❤✉♥❣ ❤➠♥❣ ✤➞② ▲❙❊ rí✐ ✤✐ ✤➳♥ ✈ị♥❣ ❦❤ỉ♥❣ ❝↕♥❤ tr❛♥❤✳ ▼ỉ ❤➻♥❤ ❧ó❝ ♥➔② ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ♥❤÷ s❛✉✿ dR = ε γ(R) − a1 RC1C − a2 RC2 dτ dC1C dτ = (−(βC2 + β0 )C1C + mC1R ) + ε[−d1C C1C + Ra1 e1 C1C ] ✭✷✳✻✮ dC 1N = ((βC2 + β0 )C1C − mC1N ) − εd1N C1N dτ dC2 = ε[−d2 C2 + a2 e2 RC2 ] − εlC2 , dτ ✈ỵ✐ d2 ❧➔ t✛ ❧➺ ❝❤➳t tü ♥❤✐➯♥ ❝õ❛ ❧♦➔✐ 2✱ d1C ✈➔ d1N ❧➔ t✛ ❧➺ ❝❤➳t tü ♥❤✐➯♥ ❝õ❛ ❧♦➔✐ t÷ì♥❣ ù♥❣ tr➯♥ ✈ị♥❣ ❝↕♥❤ tr❛♥❤ ✈➔ ✈ị♥❣ ❦❤ỉ♥❣ ❝↕♥❤ tr❛♥❤✳ ▲ó❝ ♥➔②✱ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❝↕♥❤ tr❛♥❤ ❜➜t ✤è✐ ①ù♥❣ trð t❤➔♥❤ ▼ỉ ❤➻♥❤ rót ❣å♥ d1C d2 < ♠✐♥ R∗ , a1 e1 a2 e2 dR a1 m = γ(R) − RC1 − a2 RC2 dt L(C2 ) dC1 C1 = [−(d1C m + d1N (βC2 + β0 )) + a1 e1 mR] dt L(C 2) dC2 = C [−(d + l) + a e R], 2 2 dt ✈ỵ✐ L(C2 ) = βC2 + β0 + m✳ ✼ ✭✷✳✼✮ ✭✷✳✽✮ ❇↔♥❣ ✷✳✶✿ ♣❤÷ì♥❣ ✣✐➸♠ ❝➙♥ ❜➡♥❣ ❝õ❛ ♠ỉ ❤➻♥❤ rót ❣å♥ ✭✷✳✽✮ ✈➔ ♣❤➙♥ t➼❝❤ sü ê♥ ✤à♥❤ ✤à❛ ✣✐➲✉ ❦✐➺♥ ✶✳ R2∗ < R1∗ ✶✳✶✳ R∗ < R2∗ < R1∗ ✶✳✷✳ R2∗ < R∗ < R1∗ ✶✳✸✳ R2∗ < R1∗ < R∗ ✷✳ R1∗ < R2∗ ✷✳✶✳ R∗ < R1∗ < R2∗ ✷✳✷✳ R1∗ < R∗ < R2∗ ✷✳✸✳ R1∗ < R∗∗ < R2∗ < R∗b ✷✳✹✳ R1∗ < R2∗ < R∗∗ < R∗ ✷✳✺✳ R1∗ < R2∗ < R∗ < R∗∗ ❦❤æ♥❣ ê♥ ✤à♥❤ ê♥ ✤à♥❤ (0, 0, 0)a (0, 0, 0) (R∗ , 0, 0) (0, 0, 0) (R∗ , 0, 0) (R1∗ , C1∗ , 0) (R∗ , 0, 0) (R2∗ , 0, C2∗ ) (0, 0, 0) (0, 0, 0) (R∗ , 0, 0) (0, 0, 0) (R∗ , 0, 0) (R2∗ , 0, C2∗ ) (0, 0, 0) (R∗ , 0, 0) ˆ Cˆ1 , Cˆ2 ) (R, (0, 0, 0) (R∗ , 0, 0) ˆ Cˆ1 , Cˆ2 ) (R, (R∗ , 0, 0) (R1∗ , C1∗ , 0) (R2∗ , 0, C2∗ ) (R1∗ , C1∗ , 0) (R1∗ , C1∗ , 0) (R2∗ , 0, C2∗ ) (R1∗ , C1∗ , 0) (R2∗ , 0, C2∗ ) R1∗ = (d1C m + d1N β0 )/(a1 e1 m), C1∗ = γ(R1∗ )L(0)/a1 mR1∗ , R2∗ = (d2 + l)/(a2 e2 ), C2∗ = γ(R2∗ )/a2 R2∗ , R∗∗ = R1∗ + d1N βγ(R2∗ )/(a1 e1 ma2 R2∗ ), ˆ = R2∗ , Cˆ1 = (γ(R) ˆ − a2 R ˆ Cˆ2 )L(Cˆ2 )/a1 mR, ˆ R Cˆ2 = a1 e1 m(R2∗ − R1∗ )/d1N a : (0, 0, 0) tỗ t ỗ tự s b : R1 < R∗∗ ⇔ R2∗ < R∗ ✽ t❤à ✤÷đ❝ ❣å✐ ỗ t ữủ t r tữỡ t ỳ t r ỗ t ỗ t t ỳ ữủ ố ợ ♠ët ❝↕♥❤ ♥➳✉ ❣✐ú❛ ❤❛✐ ❝→ t❤➸ t÷ì♥❣ ù♥❣ ❝â tữỡ t ợ ỗ t ữủ tứ ỹ ỗ t ▼ët sè t➼♥❤ ❝❤➜t ✤➦❝ tr÷♥❣ ♥❤÷ ❝→❝ ❝❧✐❝❦ ❝ü❝ số str ố ỗ t ữớ ỗ t ữủ ú tổ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❦❤↔♦ s→t✳ ❈❤ó♥❣ tỉ✐ s♦ s→♥❤ ❝→❝ tở t ỗ t s r tứ ổ t tr tú ỗ ợ ♠ët sè ♠ỉ ❤➻♥❤ ❤➺ t❤è♥❣ ♣❤ù❝ t↕♣ ❦❤→❝✳ ❈❤ó♥❣ tỉ✐ ❝ơ♥❣ t❤↔♦ ❧✉➟♥ ✈➲ ❝→❝ t➼♥❤ ❝❤➜t ♥➔② tø q s ổ tr túỗ ❞ü❛ tr➯♥ ❝→ t❤➸ ❈❤ó♥❣ tỉ✐ ①❡♠ ①➨t ❤➺ ✤ë♥❣ ỹ ởt s t tr tú ỗ tr♦♥❣ ♠ët ♠ỉ✐ tr÷í♥❣ ❝â ❝❤ù❛ ❤➺ ✤ë♥❣ ❧ü❝ ❝õ❛ ởt ỗ tự ọ t tú tỗ t t tr s t t ỗ r õ t ỗ tỗ t ọ tr ổ trữớ số ❝õ❛ ❝❤ó♥❣✳ ✶✳ ✿ ✣➸ ✤ì♥ ❣✐↔♥✱ ❝❤ó♥❣ tỉ✐ sû ổ trữớ ổ ữợ ọ ữủ ữ ổ trữớ ố ỗ tớ ỗ tự t ỗ æ ♠➔✉ ①❛♥❤ ❧→ ❝➙② ✤➟♠ ❤❛② ♥❤↕t t❤➸ ❤✐➺♥ ❦❤✉ ✈ü❝ ❝â ❝ä✳ ✣ë ✤➟♠ ♥❤↕t ❝õ❛ ♠➔✉ ①❛♥❤ t❤➸ ❤✐➺♥ ♠➟t ✤ë ❝õ❛ ❝ä✳ ❈→❝ æ ♠➔✉ tr➢♥❣ t❤➸ ❤✐➺♥ ❦❤✉ ✈ü❝ ❦❤æ♥❣ ❝â ❝ä✳ ✷✳ ✿ ▼é✐ ❝→ t❤➸ ✤➲✉ ❝â ❦❤↔ ♥➠♥❣ ❞✐ ❝❤✉②➸♥✱ ➠♥ t❤ù❝ ➠♥ ✈➔ s✐♥❤ s↔♥✳ ❈→❝ ❝→ t❤➸ ✤÷đ❝ ✤➦❝ tr÷♥❣ ❜ð✐ ♠ù❝ ♥➠♥❣ ❧÷đ♥❣ ❝õ❛ ❝❤ó♥❣✳ ▼ët ❝→ t❤➸ s➩ ❝❤➳t ♥➳✉ ♥➠♥❣ ❧÷đ♥❣ ❝õ❛ ❝❤ó♥❣ ❣✐↔♠ ❞➛♥ ✈➲ ✵✳ ◆➠♥❣ ❧÷đ♥❣ ❝õ❛ ❝❤ó♥❣ t➠♥❣ ❞➛♥ ❦❤✐ ➠♥ t❤ù❝ ➠♥ ✈➔ ❣✐↔♠ ❞➛♥ ❦❤✐ ❞✐ ❝❤✉②➸♥✳ ❑❤✐ ❝â ✤õ ♥➠♥❣ ❧÷đ♥❣✱ ❝→❝ ❝→ t❤➸ s➩ s✐♥❤ s↔♥✱ ❝→ t❤➸ s➩ ♠➜t ♥➠♥❣ ❧÷đ♥❣ ❦❤✐ s✐♥❤ s↔♥ ✈➔ ❝→❝ ❝→ t❤➸ ợ s s t t ũ ổ ữợ ❝→ t❤➸ ♠➭✳ ❙è ❧÷đ♥❣ ❝→ t❤➸ s✐♥❤ r❛ ♣❤ư t❤✉ë❝ ✈➔♦ t➼♥❤ ❝❤➜t ✤➦❝ tr÷♥❣ ❝õ❛ tø♥❣ ❧♦➔✐✳ ❈→❝ ❝→ t❤➸ ❧♦➔✐ ✤÷đ❝ ❣✐↔ t❤✐➳t s✐♥❤ s↔♥ ✈➔ ❞✐ é ộ ữợ ổ ọ ♥➳✉ ♠ët ❝→ t❤➸ t➻♠ t❤➜② ❜➜t ❦ý t❤ù❝ ➠♥ ♥➔♦ tr♦♥❣ ❝→❝ æ ❧➙♥ ❝➟♥✱ ♥â s➩ ❜➢t ✈➔ t❤ü❝ ❤✐➺♥ q✉→ tr➻♥❤ ➠♥✳ ◆➳✉ ❦❤æ♥❣ ❝â t❤ù❝ ➠♥ tr♦♥❣ ❝→❝ æ ❧➙♥ ❝➟♥✱ ❝→❝ ❝→ t❤➸ s➩ ❞✐ ❝❤✉②➸♥ ♠ët ❝→❝❤ ♥❣➝✉ ♥❤✐➯♥ ✤➳♥ ♠ët æ ❧➙♥ ❝➟♥✳ ✹✳ ✿ ❈❤ó♥❣ tỉ✐ ❝❤å♥ sû ❞ư♥❣ ♥➯♥ t↔♥❣ ♠ỉ ♣❤ä♥❣ ●❆▼❆ ✶✳✼ ✤➸ t❤✐➳t ❦➳ ✈➔ tr✐➸♥ ❦❤❛✐ ❝→❝ ♠ỉ ♣❤ä♥❣✳ ▼ỉ✐ tr÷í♥❣ ❈→❝ ❝→ t❤➸ ❈→❝ q✉→ tr➻♥❤ ❚❤ü❝ ❤✐➺♥ ♠æ ♣❤ä♥❣ ✶✶ ❍➻♥❤ ✸✳✸✿ ❙ü t❤❛② ✤ê✐ ❝õ❛ sè ❧÷đ♥❣ ❝→ t❤➸ ❝õ❛ tø♥❣ ❧♦➔✐✳ ❈→❝ ✤÷í♥❣ ❝♦♥❣ ♠➔✉ ✤ä✱ ①❛♥❤ ❞÷ì♥❣ ✈➔ ①❛♥❤ ❧→ ❝➙② ✤↕✐ tữỡ ự ợ tú ỗ ọ ổ ỗ t s ổ t tr túỗ ổ ỗ t❤à ❝❤♦ ❝→❝ ❤➺ ♣❤ù❝ t↕♣ ❚ø ❣â❝ ♥❤➻♥ ❝õ❛ ỗ t t t t ởt số tố ự t tr t ợ ữ trt ỗ t ữủ ✈➔ ▼❛tt❤✐❡✉ ▲❛t❛♣② tê♥❣ ❦➳t ✈➔♦ ♥➠♠ ✷✵✵✹✳ ◆❤ú♥❣ ❤➺ t❤è♥❣ ♣❤ù❝ t↕♣ ♥➔② ❝â ❝→❝ ✤➦❝ t➼♥❤ ❝❤✉♥❣ s❛✉✿ ✲ ❍➛✉ ❤➳t ❝→❝ ❤➺ t❤è♥❣ ♣❤ù❝ t↕♣ tr♦♥❣ t❤➳ ❣✐ỵ✐ t❤ü❝ ✤➲✉ ❝â ♠➟t ✤ë t♦➔♥ ❝ư❝ t❤➜♣✳ ✲ ❈→❝ ❤➺ ♣❤ù❝ t↕♣ ♥➔② ❝â ❦❤♦↔♥❣ ❝→❝❤ tr✉♥❣ ❜➻♥❤ ữớ t P ố ỗ t❤à ❧➔ t✉➙♥ t❤❡♦ ♣❤➙♥ ❜è ♠ô✿ pk ∼ k−α ợ pk ố ỗ t õ ❜➟❝ k✳ ❙è ♠ô α ❝õ❛ ♣❤➙♥ ♣❤è✐ ♠ô ♥â✐ ❝❤✉♥❣ ❧➔ ❣✐ú❛ ✈➔ 3✳ ✲ ❚➜t ❝↔ ❝→❝ ❤➺ ♣❤ù❝ t↕♣ ✤➲✉ ❝â sè ❝❧✉st❡r✐♥❣ ❝❛♦ ✈➔ ❦❤æ♥❣ ♣❤ö t❤✉ë❝ ✈➔♦ ❦➼❝❤ ❝ï ❝õ❛ ❝→❝ ❤➺ t❤è♥❣✳ ✸✳✸✳✷ ổ ỗ t tr tú ỗ ũ ữ ởt t u ỗ t❤à ✤➽❛ diskR (u) ✈ỵ✐ ❜→♥ ❦➼♥❤ R ✈➔ t➙♠ t t tr t u ởt ỗ t❤à G = (V, E) ❝õ❛ ♠ët ❤➺ s✐♥❤ t❤→✐ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉✿ t➟♣ ✤➾♥❤ ❧➔ t➟♣ ❝→❝ ❝→ t❤➸✿ V = {1, 2, · · · , n} tỗ t ỳ u v ũ ữ ú ợ ộ ữợ ổ ọ ổ t ✈ỵ✐ ♠ët ❣✐→ trà ①→❝ ✤à♥❤ ❝❤♦ R✱ ❝❤ó♥❣ t❛ õ tữỡ ự ởt ổ ỗ t tr túỗ P t ổ ỗ t ❱➻ ❤➛✉ ❤➳t ❝→❝ ❤➺ ♣❤ù❝ t↕♣ tr♦♥❣ t❤➳ ❣✐ỵ✐ t❤ü❝ ❝â sè ❝↕♥❤ t✉②➳♥ t➼♥❤ ✈ỵ✐ sè ✤➾♥❤ ♥➯♥ ❝→❝ ❤➺ t❤è♥❣ ♣❤ù❝ t↕♣ ♥➔② ❝â ♠➟t ✤ë t❤➜♣✳ ✣✐➲✉ ♥➔② ♣❤ị ❤đ♣ ✈ỵ✐ ❦➳t q✉↔ tr♦♥❣ ❝→❝ ♠ỉ ♣❤ä♥❣ ❝õ❛ ❝❤ó♥❣ tỉ✐ ✭①❡♠ ❇↔♥❣ ✸✳✷✮✳ ❍ì♥ ♥ú❛✱ ❦➳t q✉↔ t❤ü❝ ♥❣❤✐➺♠ ❝õ❛ ❝❤ó♥❣ tỉ✐ ❝❤♦ t❤➜② ❦❤♦↔♥❣ ❝→❝❤ tr ỳ t ỗ tớ số str t ữủ tứ ữợ ổ ọ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ ❦➼❝❤ ❝ï ❝õ❛ ❝→❝ ❤➺✳ ❑➳t q✉↔ ♥➔② ❝ơ♥❣ t÷ì♥❣ tü ♥❤÷ t➼♥❤ ❝❤➜t ✶✷ ❝õ❛ ❝→❝ ❤➺ ♣❤ù❝ t↕♣ ❦❤→❝ ✭①❡♠ ❇↔♥❣ ✸✳✷✮✳ ❙ü ❦❤→❝ ❜✐➺t tr♦♥❣ ❝→❝ ❦➳t q✉↔ t❤✉ ✤÷đ❝ tø ♠ỉ ❤➻♥❤ ❝õ❛ ❝❤ó♥❣ tỉ✐ ✤÷đ❝ t❤➸ ❤✐➺♥ tr♦♥❣ ❍➻♥❤ ✸✳✻✳ ✣â ố ỗ t tr ổ ❧➔ tü❛ ♣❤➙♥ ♣❤è✐ ❝❤✉➞♥✱ sè ❝→❝ ✤➾♥❤ ❝â ❜➟❝ ❝❛♦ r➜t ♥❤ä ✈➔ sè ❝→❝ ✤➾♥❤ ❝â ❜➟❝ ♥❤ä t❤➻ ❦❤ỉ♥❣ ❧ỵ♥✳ ✣✐➲✉ ♥➔② t❤➸ ❤✐➺♥ t➼♥❤ ❝♦ ❝ư♠ tr ổ s t tr tú ỗ ❜✐➺t s♦ ✈ỵ✐ ❝→❝ ❤➺ ♣❤ù❝ t↕♣ ❦❤→❝ tr♦♥❣ t❤➳ ❣✐ỵ✐ t❤ü❝✳ ◆❣♦➔✐ r❛✱ ❜➔✐ t♦→♥ t➻♠ ❝❧✐❝❦ ❝ü❝ ✤↕✐ tr ỗ t t ý t Põ õ r➜t ♥❤✐➲✉ t❤✉➟t t♦→♥ ✤➣ ✤÷đ❝ ✤÷❛ r❛ ♥❤÷♥❣ ❝❤➾ ❝â r➜t ➼t t❤✉➟t t♦→♥ ❝â t❤➸ ❧➟♣ tr➻♥❤ →♣ tr ỗ t t ú tổ ❧ü❛ ❝❤å♥ sû ❞ö♥❣ t❤✉➟t t♦→♥ t➻♠ k✲❝❧✐q✉❡ ❝õ❛ P❛❧❛ ✈➔ ❝→❝ ❝ë♥❣ sü ✤➣ ✤➲ ①✉➜t ♥➠♠ ✷✵✵✺✳ ❈→❝ ❦➳t q✉↔ tr♦♥❣ ❇↔♥❣ ✸✳✸ ✤➣ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ❤✐➺✉ q✉↔ ❝õ❛ t❤✉➟t t♦→♥ ♥➔② ✈➔ ❝✉♥❣ ❝➜♣ ❝→❝ t❤æ♥❣ t✐♥ ✈➲ t➼♥❤ ❝♦ ❝ö♠ ❝õ❛ ❝→❝ ❝→ t❤➸ ❧♦➔✐ tr♦♥❣ ♠ỉ✐ tr÷í♥❣ sè♥❣✳ ❍➻♥❤ ✸✳✺✿ ▼ỉ ❤➻♥❤ ❝→ t❤➸ tr ổ ỗ t tữỡ ự ✭❜➯♥ ♣❤↔✐✮✳ ▼ët sè ❦➳t q✉↔ tø ❝→❝ ♠æ ♣❤ä♥❣ tr tú ỗ ợ ộ ỗ t tr ộ ữợ ởt ổ ọ n m density c ✈➔ d t÷ì♥❣ ù♥❣ ❧➔ sè ✤➾♥❤✱ sè ❝↕♥❤✱ ♠➟t ✤ë✱ sè ❝❧✉st❡r✐♥❣ ✈➔ ❦❤♦↔♥❣ ❝→❝❤ tr✉♥❣ ❜➻♥❤✳ ❇↔♥❣ ✸✳✷✿ n m density c d ✻✺✸✷ ✼✶✷✸✸ ✶✳✼❡✲✸ ✵✳✻✺✸✾ ✼✳✻✾ ✻✶✶✹ ✻✼✵✸✶ ✶✳✽❡✲✸ ✵✳✻✺✷ ✼✳✽✶ ✺✹✶✷ ✻✶✻✺✷ ✷✳✶❡✲✸ ✵✳✻✹✽✷ ✼✳✻✾ ✶✸ ✸✵✸✷ ✸✸✺✼✼ ✸✳✼❡✲✸ ✵✳✻✹✽✺ ✷✳✾ ✹✶✷✻ ✸✶✹✸✺ ✶✳✽❡✲✸ ✵✳✻✽✸✸ ✹✳✸✷ ✹✺✶✹ ✸✼✸✷✵ ✶✳✽❡✲✸ ✵✳✻✼✸✼ ✹✳✼✷ ❛✮ ❜✮ ❝✮ P ố ỗ t t ởt số ữợ ởt ổ ọ ữợ ữợ ữợ ữợ ố ỗ t t ữợ ởt ổ ọ tr túỗ ♥♦✳ ♦❢ ✈❡rt✐❝❡s ♥♦✳ ♦❢ ♠❛①✐♠✉♠ ❝❧✐q✉❡ ❝❧✐q✉❡ ♥✉♠❜❡r ✹✾✵ ✻ ✺ ♥♦✳ ♦❢ ✹✲❝❧✐q✉❡ ♥♦✳ ♦❢ ✸✲❝❧✐q✉❡ ✷✹ ✽✻ ✸✳✹ ❑➳t ❧✉➟♥ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ♠ët tr♦♥❣ ♥❤ú♥❣ ❤➺ s✐♥❤ t❤→✐ ❝↕♥❤ tr❛♥❤ q✉❛♥ trồ õ tr tú ỗ ú tổ ❦➳t ❤đ♣ ♣❤÷ì♥❣ ♣❤→♣ t✐➳♣ ❝➟♥ ❞ü❛ tr➯♥ ❝→ t❤➸ t ỹ tr ỗ t tr q✉→ tr➻♥❤ ♠ỉ ❤➻♥❤ ❤â❛✳ ❈❤ó♥❣ tỉ✐ ✤➣ ❝❤➾ r❛ r➡♥❣ ✈ỵ✐ ❝→❝❤ t✐➳♣ ❝➟♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ❝â t❤➸ t t tổ t tứ ổ ỗ t ✤➸ ❤✐➸✉ s➙✉ ❤ì♥ ✈➲ ❤➺ s✐♥❤ t❤→✐ ♥➔② ❝❤➥♥❣ ❤↕♥ ♥❤÷ ❦❤↔♦ s→t ✤÷đ❝ t➼♥❤ ❝♦ ❝ư♠ ❝õ❛ ❧♦➔✐ t❤ỉ♥❣ q✉❛ t➼♥❤ ❝→❝ ❝❧✐❝❦✱ ❦❤↔♦ s→t ✤÷đ❝ ♠ù❝ ✤ë sü ♣❤➙♥ ❜è ❝õ❛ ❧♦➔✐ t❤æ♥❣ q✉❛ ❝→❝ ❦❤→✐ ♥✐➺♠ ♠➟t ✤ë ✤à❛ ♣❤÷ì♥❣✱ ♠➟t ✤ë t♦➔♥ ❝ư❝✱ ❦❤♦↔♥❣ ❝→❝❤ tr ố ỗ t ♠ỉ ♣❤ä♥❣ ✤÷đ❝ tr➻♥❤ ❜➔② ✤➸ ♠✐♥❤ ❤å❛ ❝❤♦ ❦➳t q✉↔ ❝õ❛ ❝❤ó♥❣ tỉ✐✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❞ü❛ tr➯♥ ❜➔✐ ❜→♦ ❬✹❪✱ tr♦♥❣ ❉❛♥❤ ♠ư❝ ❝→❝ ❝ỉ♥❣ tr➻♥❤ ✤➣ ❝ỉ♥❣ ❜è ❝õ❛ ❧✉➟♥ →♥✳ ❈❤÷ì♥❣ ✹ Ù◆● ❉Ư◆●✿ ▼➷ ❍➐◆❍ ❍➶❆ ▼❐❚ ❙➮ ❍➏ ❙■◆❍ ❚❍⑩■ ❈Ö ❚❍➎ ✶✹ ✹✳✶ ▼ỉ ❤➻♥❤ ❤â❛ ❤➺ ❚❤✐♦❢✲❇↕❝❤ t✉ë❝ ✹✳✶✳✶ ●✐ỵ✐ t❤✐➺✉ ❍✐➺♥ t÷đ♥❣ s✐♥❤ t❤→✐ ❝õ❛ ❧♦➔✐ ❝→ ❚❤✐♦❢ ✈➔ ❜↕❝❤ t✉ë❝ ð ❙❡♥❡❣❛❧ ❞➝♥ ❝❤ó♥❣ tỉ✐ ✤➳♥ ✈✐➺❝ ①❡♠ ①➨t✱ ♥❣❤✐➯♥ ❝ù✉ ♠ët sè ♠æ ❤➻♥❤ t♦→♥ ❤å❝ ❝õ❛ tr ũ ởt ỗ t ❝❤✉♥❣ ✈➔ ❝ò♥❣ ❝❤à✉ ♠ët →♣ ❧ü❝ ✤→♥❤ ❜➢t ❝→ ♥❤÷ ♥❤❛✉✳ ✹✳✶✳✷ ❇✐➸✉ ❞✐➵♥ ♠ỉ ❤➻♥❤ ▼ỉ ❤➻♥❤ ✶✿ tr÷í♥❣ ❤đ♣ ❦❤ỉ♥❣ ❝â ♥ì✐ tró ➞♥ dn1 n2 n1 dt = r1 n1 − K1 − a12 K1 − q1 n1 E dn n n = r2 n2 − − a21 dt K2 K2 − q2 n2 E, ✭✹✳✶✮ ✈ỵ✐ ni ❜✐➸✉ ❞✐➵♥ ♠➟t ✤ë ❝õ❛ ❧♦➔✐ i✱ i ∈ {1, 2}✳ ❈→❝ t❤❛♠ sè ri ✈➔ Ki ❧➔ t✛ ❧➺ t➠♥❣ tr÷ð♥❣ ✈➔ sù❝ ❝❤ù❛ ♠ỉ✐ tr÷í♥❣ ❝õ❛ ❧♦➔✐ i✱ i ∈ {1, 2}✳ ❚❤❛♠ sè E ❧➔ ♠ët ❤➡♥❣ sè ❜✐➸✉ ❞✐➵♥ ❤➺ sè ✤→♥❤ ❜➢t✳ qi ✤↕✐ ❞✐➺♥ ❝❤♦ t✛ ❧➺ ✤→♥❤ ❜➢t ❝õ❛ ❧♦➔✐ ❝→ i✱ i ∈ {1, 2}✳ ❱ỵ✐ ❣✐↔ t❤✐➳t ❧♦➔✐ ❧➔ ❧♦➔✐ ♠↕♥❤ ❝â ❦❤↔ ♥➠♥❣ ❦❤❛✐ t❤→❝ t❤ù❝ ➠♥ tèt ❤ì♥ ❧♦➔✐ ❧➔ ❧♦➔✐ ②➳✉✱ ❦❤✐ ①➨t ð ✤à❛ ♣❤÷ì♥❣✳ ❑❤✐ ✤â✱ ❝↕♥❤ tr❛♥❤ ❜➜t ✤è✐ ①ù♥❣ ❞➝♥ ✤➳♥ ✤✐➲✉ ❦✐➺♥ s❛✉✿ a12 K2 a21 K1 ❝â ♥❣❤➽❛ ❧♦➔✐ i ❜à ❦❤❛✐ t❤→❝ q✉→ ♠ù❝ ✈➔✴❤♦➦❝ t✛ ❧➺ tû ✈♦♥❣ ð ♥ì✐ ➞♥ ♥→✉ ❝❛♦✳ ❚r♦♥❣ ❦❤✐ ✤✐➲✉ ❦✐➺♥ Ii > ❧✐➯♥ q✉❛♥ ✤➳♥ tr÷í♥❣ ❤đ♣ ❧♦➔✐ i ❝â t❤➸ ①➙♠ ❝❤✐➳♠ ❦❤✐ sè ❧÷đ♥❣ ❧♦➔✐ ❧➔ ❤✐➳♠✱ i ∈ {1, 2}✳ ❚â♠ ❧↕✐✱ ♥➳✉ ❝→❝ ❧♦➔✐ ❜à ❦❤❛✐ t❤→❝ q✉→ ♠ù❝ ✈➔✴❤♦➦❝ t✛ ❧➺ ❝❤➳t ❝❛♦ ð ♥ì✐ tró ➞♥ t❤➻ ♥â s➩ ❜à t✉②➺t ❝❤õ♥❣✳ ✹✳✶✳✸ P❤➙♥ t➼❝❤ ✈➔ ❚❤↔♦ ❧✉➟♥ ❈→❝ ❦➳t q✉↔ q✉❛♥ trå♥❣ t❤✉ ✤÷đ❝ tø ❝→❝ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ✈➔ sü ♣❤➙♥ t➼❝❤ ê♥ ✤à♥❤ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ♠ỉ ❤➻♥❤ rót ❣å♥✳ ổ t r ữợ ởt số ❝❤♦ →♣ ❧ü❝ ✤→♥❤ ❜➢t ❝→✱ ❤➺ ✤ë♥❣ ❧ü❝ ❝õ❛ ❤❛✐ ❧♦➔✐ ❝→ ❝â t❤➸ t✐➳♥ ✤➳♥ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ê♥ ✤à♥❤ ♠➔ ð ✤➙②✱ ❧♦➔✐ ❝↕♥❤ tr❛♥❤ ②➳✉ ✭❜↕❝❤ tở ỡ s tỗ t tr tr ♠↕♥❤ ❤ì♥ ✭❚❤✐♦❢✮ s➩ ❜à t✉②➺t ❝❤õ♥❣✳ ❚r÷í♥❣ ❤đ♣ ♥➔② ①↔② r❛ ❦❤✐ O1 > ✈➔ O2 < 0✳ ◆â✐ ❝→❝❤ ❦❤→❝✱ ♠æ ❤➻♥❤ ❞ü ✤♦→♥ sü t✉②➺t ❝❤õ♥❣ ❝õ❛ ❧♦➔✐ ❝↕♥❤ tr❛♥❤ ♠↕♥❤ ❤ì♥ ①↔② r❛ ❦❤✐✿ ✲ ⑩♣ ❧ü❝ ✤→♥❤ ❜➢t ❧➔ ✤õ ❧ỵ♥ ✤➸ tè❝ ✤ë t➠♥❣ tr÷ð♥❣ t♦➔♥ ❝ư❝ A ❝õ❛ ❧♦➔✐ ❝↕♥❤ tr❛♥❤ ♠↕♥❤ ❧➔ ➙♠ ❤❛② O1 > 0✱ ❦❤✐ ✤â ❧♦➔✐ ❝↕♥❤ tr❛♥❤ ♠↕♥❤ ❤ì♥ ❜à ❦❤❛✐ t❤→❝ q✉→ ♥❤✐➲✉✳ ✲ ⑩♣ ❧ü❝ ✤→♥❤ ❜➢t ❧➔ ❧ỵ♥ ✤è✐ ✈ỵ✐ ❧♦➔✐ ❝↕♥❤ tr❛♥❤ ②➳✉ ✭❜↕❝❤ t✉ë❝✮ ♥❤÷♥❣ tè❝ ✤ë t➠♥❣ tr÷ð♥❣ t♦➔♥ ❝ư❝ ❝õ❛ ❝❤ó♥❣ M ✈➝♥ ❞÷ì♥❣✱ O2 < 0✱ ❧♦➔✐ ❝↕♥❤ tr❛♥❤ ②➳✉ ❦❤✐ ✤â ❦❤æ♥❣ ❜à ❦❤❛✐ t❤→❝ q✉→ ♥❤✐➲✉✳ ◆❣♦➔✐ r❛✱ t➻♥❤ ❤✉è♥❣ ❧♦➔✐ ❝↕♥❤ tr❛♥❤ ♠↕♥❤ ❜à t✉②➺t ❝❤õ♥❣ ❝ô♥❣ ❝â t❤➸ ①↔② r❛ ❦❤✐ ❤❛✐ t✛ ❧➺ t➠♥❣ tr÷ð♥❣ t♦➔♥ ❝ư❝ ❧➔ ❞÷ì♥❣ A > ✈➔ M > 0✳ ✣✐➲✉ ♥➔② ❝â ♥❣❤➽❛ ❧➔ →♣ ❧ü❝ ✤→♥❤ ❜➢t ❝→ ❦❤ỉ♥❣ ✤õ ❧ỵ♥ ✤➸ ❣➙② r❛ tè❝ ✤ë t➠♥❣ tr÷ð♥❣ t♦➔♥ ❝ư❝ ➙♠ t➼♥❤ ❝❤♦ ❝↔ ❚❤✐♦❢ ❧➝♥ ❜↕❝❤ t✉ë❝✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ ❤❛✐ ✤✐➲✉ ❦✐➺♥✱ I1 < ✈➔ I2 > 0✱ ❝â t❤➸ ✤÷đ❝ t t ủ ợ ữ s ✿ ✶✼ Inferior Competitor 60 40 20 0 ❍➻♥❤ ✹✳✷✿ ❝ö❝✳ 10 15 20 Superior Competitor 25 30 ❱➼ ❞ư ♠✐♥❤ ❤å❛ ❝❤♦ ♠ỉ ❤➻♥❤ ✷ ❦❤✐ ❧♦➔✐ ❝↕♥❤ tr❛♥❤ ②➳✉ ❤ì♥ t❤➢♥❣ t♦➔♥ 120 Inferior Competitor 100 80 60 40 20 0 10 20 30 Superior Competitor ❍➻♥❤ ✹✳✹✿ ❝ư❝✳ ❱➼ ❞ư ♠✐♥❤ ❤å❛ ❝❤♦ ♠ỉ ❤➻♥❤ ✸ ❦❤✐ ❧♦➔✐ ❝↕♥❤ tr❛♥❤ ②➳✉ ❤ì♥ t❤➢♥❣ t♦➔♥ I1 < (4.2) tữỡ ữỡ ợ K1 × K2 q1 E d1 ν2∗ − r1 r1 ν1∗ q2 E q2 µ∗2 1− − r2 r2 µ∗1 1− ợ à1 = m/(0 + m), à2 = µ∗1 ✳ ✲ I2 > ✈➔ (4.2) t÷ì♥❣ ✤÷ì♥❣ ✈ỵ✐ q2 E 1− − r2 K2 K2 < a21 < × K1 K1 q1 E 1− − r1 ✶✽ < a12 < q2 µ∗2 r2 µ∗1 d1 ν2∗ r1 ν1∗ K1 , K2 − αd2 K1 mr2 ◆❤ú♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ♥➔② ❝❤♦ t❤➜② t→❝ ✤ë♥❣ t✐➯✉ ❝ü❝ ❝õ❛ sü ❝↕♥❤ tr❛♥❤ ố ợ sỹ t tr t tữỡ ự ❜↕❝❤ t✉ë❝✮ ❣➙② r❛ ❜ð✐ ❜↕❝❤ t✉ë❝ ✭t÷ì♥❣ ù♥❣ ❚❤✐♦❢✮ ợ tữỡ ự ọ ỳ tr ợ ữủ tứ ✶ ❧➔ ✤è✐ t❤õ ❝↕♥❤ tr❛♥❤ ♠↕♥❤ ♥➳✉ ❦❤æ♥❣ ❝â sü ✤→♥❤ ❜➢t ❝→✳ ❚✉② ♥❤✐➯♥✱ ❞♦ ♠ët sè ❧÷đ♥❣ ❧ỵ♥ ❝→❝ t❤❛♠ sè✱ ❝â t❤➸ ❧ü❛ ❝❤å♥ ❝→❝ t❤❛♠ sè ✤➸ ❦✐➸♠ tr❛ t➜t ❝↔ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝✳ ❚r♦♥❣ ❤❛✐ tr÷í♥❣ ❤đ♣ ❞➝♥ ✤➳♥ sü t✉②➺t ❝❤õ♥❣ ❝õ❛ ❧♦➔✐ ❝↕♥❤ tr❛♥❤ ♠↕♥❤✱ ❝❤ó♥❣ tỉ✐ t✐♥ r➡♥❣ tr÷í♥❣ ❤đ♣ ✤➛✉ t✐➯♥ ✭A < ✈➔ M > 0✮ ❧➔ ♠ët ❧ü❛ ❝❤å♥ tèt ✤➸ ❣✐↔✐ t❤➼❝❤ ♥❤ú♥❣ q✉❛♥ s→t ữủ tỹ ố ợ tở ❙❡♥❡❣❛❧✳ ✹✳✷ ▼ỉ ❤➻♥❤ ❤â❛ ❤➺ ▲ó❛✲❘➛② ♥➙✉ ✹✳✷✳✶ ●✐ỵ✐ t❤✐➺✉ ❘➛② ♥➙✉ ✤➣ ✤÷đ❝ ❣❤✐ ♥❤➟♥ ❧➔ ❧♦➔✐ ❣➙② q trồ t t ữợ trỗ ❧ó❛✳ ❈→❝ ✤➦❝ t➼♥❤ s✐♥❤ ❤å❝ ✈➔ s✐♥❤ t❤→✐ ❝õ❛ r➛② ♥➙✉ ✈➔ ❧ó❛ ✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ♥❤✐➲✉ ♥➠♠ ♥❤÷♥❣ ♠ỉ ❤➻♥❤ t♦→♥ ❤å❝ ✤➣ ❦❤ỉ♥❣ ✤÷đ❝ sû ❞ư♥❣ ✤➸ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❤✐➺♥ t÷đ♥❣ s✐♥❤ t❤→✐ ♥➔②✳ ✣✐➲✉ ✤â ❞➝♥ ✤➳♥ ♠ö❝ ✤➼❝❤ ❝õ❛ ♥❣❤✐➯♥ ❝ù✉ ♥➔② ❧➔ ①➙② ❞ü♥❣ ♠ët ♠æ ❤➻♥❤ t♦→♥ ❤å❝ ♠æ t↔ ❤➺ s✐♥❤ t❤→✐ r➛② ♥➙✉✲❧ó❛✳ ✹✳✷✳✷ ▼ỉ ❤➻♥❤ ❤â❛ ❈❤ó♥❣ tỉ✐ ①➨t ❤➺ s✐♥❤ t❤→✐ r➛② ♥➙✉✲❧ó❛ ♥❤÷ ♠ët ❤➺ tr túỗ tr ổ trữớ ũ ú tổ ❣✐↔ t❤✐➳t ❧ó❛ t➠♥❣ tr÷ð♥❣ t❤❡♦ ❞↕♥❣ ❧♦❣✐st✐❝ tr➯♥ ❝↔ ❤❛✐ ✈ị♥❣✳ ❍ì♥ ♥ú❛✱ ❦❤✐ r➛② ♥➙✉ ➠♥ ❧ó❛✱ ♠➟t ✤ë ❝õ❛ ❧ó❛ s➩ ❣✐↔♠ tr♦♥❣ ❦❤✐ ♠➟t ✤ë ❝õ❛ r➛② ♥➙✉ s➩ t➠♥❣✳ ❘➛② ♥➙✉ s➩ ❜à tr✐➺t t✐➯✉ ❦❤✐ ❦❤ỉ♥❣ ❝â ❧ó❛ tr➯♥ ❝↔ ❤❛✐ ✈ị♥❣✳ ❈→❝ ❝→ t❤➸ r➛② ♥➙✉ ✤÷đ❝ ❣✐↔ t❤✐➳t ❝â t❤➸ ❞✐ ❝❤✉②➸♥ ❣✐ú❛ ❤❛✐ ✈ị♥❣ ✈➔ q✉→ tr➻♥❤ ❞✐ ❝÷ ♥➔② ❞✐➵♥ r❛ tr♦♥❣ t❤❛♥❣ t❤í✐ ❣✐❛♥ ♥❤❛♥❤ ❤ì♥ s♦ ✈ỵ✐ q✉→ tr➻♥❤ t➠♥❣ tr÷ð♥❣✱ q✉→ tr➻♥❤ ❝↕♥❤ tr❛♥❤ ❤❛② q✉→ tr➻♥❤ ú ỵ ni t ú t÷ì♥❣ ù♥❣ tr➯♥ ✈ị♥❣ i✱ i ∈ {1, 2}✳ piA ✱ piJ ❧➔ ♠➟t ✤ë ❝õ❛ ❧♦➔✐ r➛② ♥➙✉ ð ❣✐❛✐ ✤♦↕♥ tr÷ð♥❣ t❤➔♥❤ ✈➔ ✈➔ ð ❣✐❛✐ ✤♦↕♥ trù♥❣ t÷ì♥❣ ù♥❣ tr➯♥ ✈ị♥❣ i✱ i ∈ {1, 2}✳ ❈→❝ t❤❛♠ sè r1 ✱ r2 ✈➔ K ❧➔ tè❝ ✤ë t➠♥❣ tr÷ð♥❣ ✈➔ sù❝ ❝❤ù❛ ♠ỉ✐ tr÷í♥❣ ❝õ❛ ❧ó❛ t÷ì♥❣ ù♥❣ tr➯♥ ♠é✐ ✈ò♥❣✳ ❚❤❛♠ sè diA , diJ ❜✐➸✉ ❞✐➵♥ t✛ ❧➺ ❝❤➳t tü ♥❤✐➯♥ ❝õ❛ r➛② ♥➙✉ ð ❣✐❛✐ ✤♦↕♥ tr÷ð♥❣ t❤➔♥❤ ✈➔ r➛② ♥➙✉ ð ❣✐❛✐ ✤♦↕♥ trù♥❣ t÷ì♥❣ ù♥❣ tr➯♥ ✈ị♥❣ i✱ i ∈ {1, 2}✳ ❜✐➸✉ ❞✐➵♥ t✛ ❧➺ ➠♥ ❧ó❛ ❝õ❛ r➛② ♥➙✉✱ ei ❧➔ t❤❛♠ sè ❧✐➯♥ q✉❛♥ ✤➳♥ sü ❝❤✉②➸♥ ❤â❛ t❤ù❝ ➠♥ t÷ì♥❣ ù♥❣ tr➯♥ ✈ị♥❣ i✱ i ∈ {1, 2}✳ ❈❤ó♥❣ tỉ✐ ❣✐↔ t❤✐➳t m, m ❧➔ t✛ ❧➺ ❞✐ ❝÷ ❝õ❛ r➛② ♥➙✉ ð ❣✐❛✐ ✤♦↕♥ tr÷ð♥❣ t❤➔♥❤ tø ✈ị♥❣ ✤➳♥ ✈ị♥❣ ✈➔ ♥❣÷đ❝ ❧↕✐✳ ❚❤❛♠ sè ε ❜✐➸✉ ❞✐➵♥ t✛ ❧➺ ❣✐ú❛ ❤❛✐ t❤❛♥❣ t❤í✐ ❣✐❛♥ t = ετ ✳ ❱➻ ✤➙② ❧➔ ❜➔✐ t♦→♥ s✐♥❤ t❤→✐ ♥➯♥ ❝❤ó♥❣ tỉ✐ ❝❤➾ ①➨t ♥❣❤✐➯♠ (n1 , n2 , p1A , p2A , p2J ) ✈ỵ✐ ❝→❝ ❣✐→ trà ❜❛♥ ✤➛✉ ❦❤æ♥❣ ➙♠ n1 (0) ≥ 0, n2 (0) ≥ 0, p1A (0) ≥ 0, p2A (0) ≥ 0, p1J (0) ≥ 0, p2J (0) ≥ 0✳ ▼ỉ ❤➻♥❤ ✤➛② ✤õ t❤✉ ✤÷đ❝ ♥❤÷ s❛✉✿ ✶✾ n1 dn1 dτ = ε r1 n1 − K − a1 n1 p1A n2 dn2 = ε r2 n2 − − a2 n2 p2A dτ K dp1A = ε − d1A p1A + α1 p1J + mp2A − mp1A dτ dp2A = ε − d2A p2A + α2 p2J + mp1A − mp2A dτ dp1J = ε − d1J p1J − α1 p1J + e1 a1 n1 p1A dτ dp2J = ε − d p − α p + e a n p 2J 2J 2J 2 2A d P t rữợ t t➼♥❤ ❝❤➜t ❦❤æ♥❣ ➙♠ ✈➔ ❜à ❝❤➦♥ ❝õ❛ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ❤♦➔♥ t❤✐➺♥ ❜❛♥ ✤➛✉ ✤÷đ❝ ❝❤➾ r❛✳ ❚✐➳♣ t❤❡♦✱ sỷ ữợ ữỡ tờ ❤đ♣ ❜✐➳♥✱ ♠ỉ ❤➻♥❤ rót ❣å♥ ♥❤➟♥ ✤÷đ❝ ❧➔✿ ▼ỉ ❤➻♥❤ rót ❣å♥ dn1 n1 dt = n1 r1 − K − a1 µ1 pA dn2 n2 − a2 µ2 pA = n2 r2 − dt K dpA = − d1A µ1 + d2A µ2 pA + α1 p1J + α2 p2J dt dp1J = −d1J p1J − α1 p1J + e1 a1 n1 µ1 pA dt dp2J = −d p − α p + e a n µ p 2J 2J 2J 2 2 A dt m m ✈ỵ✐ µ1 = ✱ µ2 = ✈➔ pA = p1A + p2A ✳ m+m m+m ✷✵ P❤➙♥ t➼❝❤ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ♠ỉ ❤➻♥❤ rót ❣å♥ ❚❛ t❤➜② ❧✉ỉ♥ ❝â ✸ ✤✐➸♠ ❝➙♥ ❜➡♥❣ s❛✉ tr➯♥ trö❝ tå❛ ✤ë✿ E0 (0, 0, 0, 0, 0)✱ E1 (0, K, 0, 0, 0)✱ ✈➔ E2 (K, 0, 0, 0, 0)✳ ✣✐➸♠ ❝➙♥ ❜➡♥❣ E3 (K, K, 0, 0, 0) ổ tỗ t ❜➡♥❣ E4 (nˆ1 , 0, pˆA , pˆ1J , 0) tỗ t ữợ 1/K < 1/ n1 ợ n ˆ1 = r1 − nˆK1 (d1A µ1 + d2A µ2 )(d1J + α1 ) (d1A µ1 + d2A µ2 )ˆ pA , pˆA = , pˆ1J = , µ1 e1 a1 α1 a1 µ1 α1 ✣✐➸♠ E5 (0, n , pA , 0, p2J ) tỗ t↕✐ ♥➳✉ 1/K < 1/¯ n2 ✈ỵ✐ n ¯2 = r2 − n¯K2 (d1A µ1 + d2A µ2 )(d2J + α2 ) (d1A µ1 + d2A µ2 )¯ pA , p¯A = , p¯2J = µ2 e2 a2 α2 a2 µ2 α2 ✣✐➸♠ ❝➙♥ ❜➡♥❣ ❝✉è✐ ❝ị♥❣ ❧➔ ♠ët ✤✐➸♠ tr♦♥❣ E6 (n∗1 , n∗2 , p∗A , p1J , p2J ) tỗ t ❦✐➺♥ s❛✉ ✤➙② t❤ä❛ ♠➣♥✿ max{ nˆ11 , n¯12 } < K < nˆ11 + n¯12 ✱ ✈ỵ✐ α1 e1 a1 µ1 α2 e2 a2 µ2 d1A µ1 + d2A µ2 + − d + α d + α K 1J 2J , p∗A = a1 µ1 α1 e1 a1 à1 a2 à2 e2 a2 à2 ì + × r d1J + α1 r2 d2J + α2 a1 µ1 ∗ a2 µ2 ∗ ∗ ∗ n1 = K − pA , n = K − pA , r1 r1 e1 a1 µ1 e2 a2 µ2 p∗1J = n∗1 p∗A , p∗2J = n∗2 p∗A (d1J + α1 ) (d2J + α2 ) ỡ ỗ t õ t ữủ ❧♦↕✐ t❤❡♦ K ✳ ❚❛ ❝â t❤➸ ❞➵ ❞➔♥❣ t❤➜② 1 ♥➳✉ K > nˆ + n¯ t❤➻ ❤➺ ✤ë♥❣ ❧ü❝ ❝â ✤✐➸♠ ❝➙♥ ❜➡♥❣✿ E0 , E1 , E2 , E3 ✈➔ ✤✐➸♠ ❝➙♥ ❜➡♥❣ E3 ❧➔ ❧✉æ♥ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥✳ ◆➳✉ ✤✐➸♠ E6 tỗ t t E4 , E5 ✤➲✉ ❦❤æ♥❣ ê♥ ✤à♥❤✳ ❱➔ ❝→❝ ✤✐➸♠ ❝➙♥ ❜➡♥❣ E4 , E5 ❦❤ỉ♥❣ t❤➸ ❝ị♥❣ ê♥ ✤à♥❤✳ ❈â t❤➸ ♣❤➙♥ t➼❝❤ ❝❤✐ t✐➳t t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ❤➺ ♥❤÷♥❣ ❦❤ỉ♥❣ ❞➵ ❞➔♥❣ ✤➸ tê♥❣ ❦➳t ❣å♥ ❧↕✐ ✈➻ ❝â r➜t ♥❤✐➲✉ t❤❛♠ sè tr♦♥❣ ♠ỉ ❤➻♥❤✳ ❈❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② tr♦♥❣ ❍➻♥❤ ✹✳✶✵ ♠ët ✈➼ ❞ư ♠➔ ð ✤â t❛ ❝â t❤➸ t❤➜② ✤÷đ❝ t♦➔♥ ❜ë ❜ù❝ tr❛♥❤ ê♥ ✤à♥❤ ❝õ❛ ♠ỉ ❤➻♥❤ ♣❤ư t❤✉ë❝ t❤❡♦ K tr♦♥❣ ❦❤✐ t➜t ❝↔ ❝→❝ t❤❛♠ sè ❝á♥ ❧↕✐ ✤➲✉ ✤÷đ❝ ❝è ✤à♥❤✳ ❈ư t❤➸✱ ❝→❝ t❤❛♠ sè ✤÷đ❝ ❝❤å♥ ♥❤÷ s❛✉✿ r1 = 0.3✱ r2 = 0.9✱ a1 = 0.7✱ a2 = 0.1✱ e1 = 0.9✱ e2 = 0.5✱ α1 = 0.1✱ α2 = 0.1✱ m = 0.8✱ m ¯ = 0.2 ✈➔ K10 ❧➔ ♥❣❤✐➺♠ ❞÷ì♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✉ t❤❡♦ ❜✐➳♥ X r22 n ¯ X + r2 v + (d1A µ1 + d2A µ2 )(d2J + α2 ) e2 α2 a2 µ2 r2 X = 0, v v ợ v = d1A à1 + d2A µ2 + d2J + α2 ✳ E6 ✭✯✮ ♥❣❤➽❛ ❧➔ ✤✐➸♠ ❝➙♥ ❜➡♥❣ E6 ê♥ ✤à♥❤ ❦❤✐ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ê♥ ✤à♥❤ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ♥â ✤÷đ❝ t❤ä❛ ♠➣♥✳ ❱➔ tr♦♥❣ tr÷í♥❣ ❤đ♣ ✭✯✯✮✱ ❦➳t q✉↔ ❝õ❛ ❝→❝ ♠æ ♣❤ä♥❣ ❝❤♦ t❤➜② r➡♥❣ ❤➺ ✤ë♥❣ ❧ü❝ ❝â ♠ët ❝❤✉ tr➻♥❤ ❣✐ỵ✐ ❤↕♥ q✉❛♥❤ tr↕♥❣ t❤→✐ ❝➙♥ ❜➡♥❣ E5 ✳ ✷✶ ❙♦ s→♥❤ ♠➟t ✤ë ❧ó❛ tr➯♥ ✈ị♥❣ ❣✐ú❛ ♠ỉ ❤➻♥❤ ✤➛② ✤õ ✈➔ ♠ỉ ❤➻♥❤ rót ❣å♥✳ ❚r÷í♥❣ ❤đ♣✿ r➛② ♥➙✉ tr✐➺t t✐➯✉✱ ❝❤➾ ❝á♥ ❧ó❛ ♣❤→t tr✐➸♥ tr➯♥ ❝↔ ❤❛✐ ✈ò♥❣✳ ❍➻♥❤ ✹✳✾✿ ❍➻♥❤ ✹✳✶✵✿ ❈→❝ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ✈➔ ♣❤➙♥ t➼❝❤ ê♥ ✤à♥❤ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ♠ỉ ❤➻♥❤ rót ❣å♥✳ P❤➙♥ t➼❝❤ t♦➔♥ ❝ư❝ ❝õ❛ ổ rút ỵ tt r a1 µ1 e1 Kα1 a2 µ2 e2 Kα2 + ≤ d1A µ1 + d2A µ2 d1J + α1 d2J + α2 ❑❤✐ ✤â ✤✐➸♠ ❝➙♥ ❜➡♥❣ E3 (K, K, 0, 0, 0) ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ t♦➔♥ ❝ö❝ tr (0, +)5 ú ỵ t ữỡ E3 ố ợ ❦✐➺♥ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ t♦➔♥ ❝ö❝ ❝õ❛ ♥â✳ r2 K n1 > ợ n1 pA ữ ●✐↔ t❤✐➳t r➡♥❣ pA > pˆA > a2 µ2 s❛✉ n1 ≤ lim inf t→∞ n1 (t) ✈➔ pA ≤ lim inf t→∞ pA (t)✳ ❑❤✐ ✤â ✤✐➸♠ ❝➙♥ ❜➡♥❣ E4 (nˆ1 , 0, pˆA , pˆ1J , 0) ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ t♦➔♥ ❝ö❝ tr➯♥ (0, +∞)5 ✳ K K ●✐↔ t❤✐➳t r➡♥❣ n1 > ✈➔ n2 > ợ n1 n2 ữ s 2 n1 ≤ lim inf t→∞ n1 (t) ✈➔ n2 ≤ lim inf t n2 (t) ỵ ỵ ✷✷ ❍➻♥❤ ✹✳✶✶✿ ❚r÷í♥❣ ❤đ♣ r➛② ♥➙✉ tr✐➺t t✐➯✉✱ ❧ó❛ ♣❤→t tr✐➸♥ tr➯♥ ❝↔ ❤❛✐ ✈ò♥❣✳ ❑❤✐ ✤â ✤✐➸♠ ❝➙♥ ❜➡♥❣ E6 (n∗1 , n∗2 , p∗A , p∗1J , p∗2J ) ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ t♦➔♥ ❝ö❝ tr➯♥ (0, +)5 ú ỵ t✐➺♠ ❝➟♥ t♦➔♥ ❝ư❝ ❝õ❛ E4 ✭t÷ì♥❣ ù♥❣ E6 ✮ ❧➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ E4 ✭t÷ì♥❣ ù♥❣ E6 ✮✳ ❈❤ó♥❣ tỉ✐ q✉❛♥ t t q ỵ tr õ r➛② ♥➙✉ ❜à t✉②➺t ❝❤õ♥❣✳ ❍✐➸✉ ✤÷đ❝ ❝ì ❝❤➳ ✤➡♥❣ s❛✉ ❦➳t q✉↔ ♥➔② ✤â♥❣ ♠ët ✈❛✐ trá q✉❛♥ trå♥❣✱ ❣✐ó♣ ❝❤♦ ❝♦♥ ♥❣÷í✐ ❝â ❝→❝ t→❝ ✤ë♥❣ ♣❤ị ❤đ♣ ✈➔♦ ❤➺ s✐♥❤ t❤→✐ r➛② ♥➙✉✲❧ó❛✳ ❈❤ó♥❣ tỉ✐ t➟♣ tr✉♥❣ ♥❣❤✐➯♥ ❝ù✉ t→❝ ✤ë♥❣ ❝õ❛ ❝→❝ ②➳✉ tè ❦❤→❝ ♥❤❛✉ ❞➝♥ ✤➳♥ sü t✉②➺t ❝❤õ♥❣ ❝õ❛ r➛② ♥➙✉✿ ❝→❝ t❤❛♠ sè ❞✐ ❝÷ ✈➔ ❝→❝ t❤❛♠ sè ✈➲ ❝➜✉ tró❝ t✉ê✐✳ ❈→❝ ♠ỉ ♣❤ä♥❣ ✤÷đ❝ tr➻♥❤ ❜➔② ✤➸ ♠✐♥❤ ❤å❛ ❝❤♦ ❦➳t q✉↔ ❝õ❛ ❝❤ó♥❣ tỉ✐ ✭①❡♠ ❍➻♥❤ ✹✳✶✶✮✳ ❑➳t ❧✉➟♥ ❈❤ó♥❣ tỉ✐ ✤➣ tr➻♥❤ ❜➔② tr♦♥❣ ❝❤÷ì♥❣ ♥➔② ♠ët sè ♠ỉ ❤➻♥❤ ❝❤♦ ❤❛✐ ❤✐➺♥ t÷đ♥❣ s✐♥❤ t❤→✐✳ ✣è✐ ✈ỵ✐ ❤➺ s✐♥❤ t❤→✐ ❤❛✐ ❧♦➔✐ ❝→ ❚❤✐♦❢✲❜↕❝❤ t✉ë❝ ð ❜í ❜✐➸♥ ❙❡♥❡❣❛❧✱ ❝❤ó♥❣ tỉ✐ ✤➣ ✤➲ ①✉➜t ❜❛ ♠ỉ tữỡ ự ợ trữớ ủ õ ự t↕♣ t➠♥❣ ❞➛♥✿ ✭✶✮ tr÷í♥❣ ❤đ♣ ❦❤ỉ♥❣ ❝â ♥ì✐ tró ➞♥✱ ✭✷✮ tr÷í♥❣ ❤đ♣ ❞✐ ❝÷ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ♠➟t ✤ë ✈➔ ✭✸✮ tr÷í♥❣ ❤đ♣ ❞✐ ❝÷ ♣❤ư t❤✉ë❝ ✈➔♦ ♠➟t ✤ë✳ ✣è✐ ✈ỵ✐ ❤➺ s✐♥❤ t❤→✐ r➛② ♥➙✉✲❧ó❛✱ ♠ët ♠ỉ ❤➻♥❤ ✤÷đ❝ ✤÷❛ r❛ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♠ỉ ❤➻♥❤ ❤â❛ ❞ü❛ tr➯♥ ♣❤÷ì♥❣ tr➻♥❤✳ ❈❤ó♥❣ tỉ✐ ✤➣ ♣❤➙♥ t➼❝❤ sü ê♥ ✤à♥❤ ✤à❛ ♣❤÷ì♥❣ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t t♦➔♥ ❝ư❝ ❝õ❛ ♠ỉ ❤➻♥❤ rót ❣å♥ ✤➸ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ♠ỉ ❤➻♥❤ ❤♦➔♥ t❤✐➺♥ ❜❛♥ ✤➛✉ ❝õ❛ ❝→❝ ❤✐➺♥ t÷đ♥❣ s✐♥❤ t❤→✐ ♥➔②✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❞ü❛ tr➯♥ ❜➔✐ ❜→♦ ❬✷❪ ✈➔ ❬✸❪✱ tr♦♥❣ ❉❛♥❤ ♠ư❝ ❝→❝ ❝ỉ♥❣ tr➻♥❤ ✤➣ ❝æ♥❣ ❜è ❝õ❛ ❧✉➟♥ →♥✳ ✷✸ ❑➌❚ ▲❯❾◆ ◆❤ú♥❣ ❦➳t q✉↔ ❝❤➼♥❤ ❧✉➟♥ →♥ ✤↕t ✤÷đ❝ ❧➔✿ ▲✉➟♥ →♥ ✤➣ ♣❤→t tr✐➸♥ ♠ët sè ♠ỉ ❤➻♥❤ ❧✐➯♥ tư❝ ✈➔ rí✐ r↕❝ ✤➸ ♥❣❤✐➯♥ ❝ù✉ t→❝ ✤ë♥❣ ❝õ❛ ♠ỉ✐ tr÷í♥❣✱ ❝→❝ ✤➦❝ t➼♥❤ ✈➲ ❤➔♥❤ ✈✐ ❝õ❛ ❝→❝ ❝→ t❤➸ ✈➔ ❝➜✉ tró❝ t✉ê✐ ❝õ❛ q✉➛♥ t❤➸ ❧♦➔✐ ✤è✐ ✈ỵ✐ ❝→❝ ❤➺ s✐♥❤ t❤→✐ ❝â ②➳✉ tè ❝↕♥❤ tr❛♥❤ t ỵ tt ự tỹ t ❈→❝ ❦➳t q✉↔ ❝ư t❤➸ ✤÷đ❝ ✤÷❛ r❛ ♥❤÷ s❛✉✿ t ỵ tt ổ ợ ❧÷đ❝ ❦❤→❝ ♥❤❛✉ ✭❝❤✐➳♥ ❧÷đ❝ ❤✉♥❣ ❤➠♥❣ ✈➔ trè♥ tr→♥❤✮ ❞ü❛ tr➯♥ sü ❞✐ ❝÷ ❝õ❛ ❝→❝ ❝→ t❤➸ tr♦♥❣ ♠ỉ✐ tr÷í♥❣ ♥❤✐➲✉ ✈ị♥❣ ❝â t➼♥❤ ❝❤➜t s✐♥❤ ❤å❝ ❤♦➦❝ ❦❤æ♥❣✱ ✤➣ ❝❤♦ t❤➜② r➡♥❣ tr♦♥❣ ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ ♥❤➜t ✤à♥❤✱ t➼♥❤ ❤✉♥❣ ❤➠♥❣ ❝â ❤✐➺✉ q✉↔ ✤è✐ ✈ỵ✐ sü sè♥❣ ❝á♥ ❝õ❛ ▲■❊ ✈➔ t❤➟♠ ❝❤➼ ❣➙② r❛ sü t✉②➺t ❝❤õ♥❣ ❝õ❛ ▲❙❊✳ ✲ P❤→t tr✐➸♥ ♣❤÷ì♥❣ ♣❤→♣ ①➙② ỹ ổ ỗ t tứ ổ ❝→ t❤➸✳ ❙♦ s→♥❤ ❝→❝ ✤➦❝ tr÷♥❣ ❝õ❛ ♠ỉ ❤➻♥❤ ỗ t ữủ ợ trữ ởt số ỗ t tố ự t tữớ ❣➦♣✳ ❱➲ ♠➦t ù♥❣ ❞ư♥❣✿ ✲ ▼ỉ ❤➻♥❤ ❝↕♥❤ tr❛♥❤ ❣✐ú❛ r➛② ♥➙✉ ✈➔ ❧ó❛ ✈ỵ✐ ❝➜✉ tró❝ t✉ê✐ ❝❤♦ ❧♦➔✐ r➛② ♥➙✉ ❝❤♦ t❤➜② ♠ët sè ❦➳t q✉↔ ♥ê✐ ❧➯♥ ❣✐ó♣ ❤é trđ q✉②➳t ✤à♥❤ ❝❤♦ ❝→❝ ♥❤➔ q✉↔♥ ỵ ổ tr ỳ tở ữợ ỹ t ✤➣ ❝❤➾ r❛ r➡♥❣✱ sü ❣✐❛ t➠♥❣ ❝õ❛ →♣ ❧ü❝ ✤→♥❤ ❜➢t ð ♠ët sè ❦❤✉ ✈ü❝ ❞➝♥ ✤➳♥ sü ❝↕♥ ❦✐➺t ❝õ❛ ❧♦➔✐ ❝→ ❚❤✐♦❢ ✈➔ ❧➔♠ t➠♥❣ ♠↕♥❤ ♠➩ sè ❧÷đ♥❣ ✤è✐ t❤õ ❝↕♥❤ tr❛♥❤ ❝õ❛ ♥â ❧➔ ❜↕❝❤ t✉ë❝✳ ▲✉➟♥ →♥ ❝â t❤➸ t✐➳♣ tö❝ t❤❡♦ ♠ët số s õ rt ữợ t ♠➔ ❝❤ó♥❣ tỉ✐ ❝â t❤➸ t✐➳♣ tư❝ ♥❣❤✐➯♥ ❝ù✉ ✤➸ ❝↔✐ t❤✐➺♥ ❦➳t q✉↔ tr♦♥❣ ❧✉➟♥ →♥ ♥➔②✳ ✲ ❚r♦♥❣ ❝→❝ ♠æ ❤➻♥❤ ❤✐➺♥ t↕✐✱ ❝→❝ ✤➦❝ t➼♥❤ ✈➲ ❤➔♥❤ ợ ữủ t tr ♠ỉ✐ tr÷í♥❣ ❝â ♠ët ✈ị♥❣ ❝↕♥❤ tr❛♥❤ ❞✉② ♥❤➜t ✈➔ ❝→❝ ✈ị♥❣ ❦❤ỉ♥❣ ❝â sü ❝↕♥❤ tr❛♥❤✳ ❙➩ ❧➔ t❤ó ✈à ❤ì♥ ❦❤✐ ①❡♠ ①➨t ♠ët sè ✈ị♥❣ ❝↕♥❤ tr❛♥❤ ✤÷đ❝ ❦➳t ♥è✐ ❜ð✐ ❞✐ ❝÷✳ ✣✐➲✉ ✤â ❝â t❤➸ ❞➝♥ ✤➳♥ ♠ỉ ❤➻♥❤ ♣❤ù❝ t↕♣ ❤ì♥ ♥❤÷♥❣ t❤ó ✈à ✤➸ ♥❣❤✐➯♥ ❝ù✉✳ ✲ ❚r♦♥❣ ♠ỉ ❤➻♥❤ rí✐ r↕❝✱ ❧✉➟♥ →♥ ♠ỵ✐ ❝❤➾ ①➨t ✤➳♥ sü ❝↕♥❤ tr❛♥❤ ❝õ❛ ❤❛✐ tú ỗ tr ởt ổ trữớ t t ✤â✱ ❝â t❤➸ ♠ð rë♥❣ ♥❣❤✐➯♥ ❝ù✉ s❛♥❣ tr÷í♥❣ ❤đ♣ ♣❤ù❝ t↕♣ ❤ì♥ ♥❤÷ ❤➺ ❝↕♥❤ tr❛♥❤ ❝â ♥❤✐➲✉ ❤ì♥ ữ ởt s t ỗ ởt ỗ tú õ t ✤÷❛ t❤➯♠ ❝→❝ ✤➦❝ t➼♥❤ ✈➲ ❞✐ ❝÷ ❝õ❛ ❝→❝ ❧♦➔✐ ❤❛② sü ❜✐➳♥ ✤ê✐ ❝õ❛ ♠ỉ✐ tr÷í♥❣ ✈➔♦ ♠ỉ ❤➻♥❤ rí✐ r↕❝✳ ✲ ❚r♦♥❣ ♠ỉ ❤➻♥❤ ❝↕♥❤ tr❛♥❤ ❝õ❛ ❤➺ s✐♥❤ t❤→✐ r➛② ♥➙✉ ✈➔ ❧ó❛ ❤✐➺♥ t↕✐ ❝❤➾ t➼♥❤ ✤➳♥ sü ❞✐ ❝÷ ✤ë❝ ❧➟♣ ♠➟t ✤ë✳ ❉♦ ✤â ❝â t❤➸ ①❡♠ ①➨t ✤➳♥ tr÷í♥❣ ❤đ♣ ❞✐ ❝÷ ♣❤ư t❤✉ë❝ ✈➔♦ ♠➟t ✤ë tr♦♥❣ ♠ỉ ❤➻♥❤ tr♦♥❣ t÷ì♥❣ ❧❛✐✳ ❉❆◆❍ ▼Ö❈ ❈⑩❈ ❈➷◆● ❚❘➐◆❍ ✣❶ ❈➷◆● ❇➮ ❈Õ❆ ▲❯❾◆ ⑩◆ ✶✳ ❚❤✉② ◆❣✉②❡♥✲P❤✉♦♥❣✱ ❉♦❛♥❤ ◆❣✉②❡♥✲◆❣♦❝ ✭✷✵✶✻✮ ❊❢❢❡❝ts ♦❢ ❇❡❤❛✈✐♦✉r❛❧ ❙tr❛t❡❣② ♦♥ t❤❡ ❊①♣❧♦✐t❛t✐✈❡ ❈♦♠♣❡t✐t✐♦♥ ❉②♥❛♠✐❝s✳ ❆❝t❛ ❇✐♦t❤❡♦r❡t✐❝❛✱ ✻✹✱ ♣♣✳ ✹✾✺✲✺✶✼✳ ✭❙❈■✮ ✷✳ ❚❤✉② ◆❣✉②❡♥✲P❤✉♦♥❣✱ ❉♦❛♥❤ ◆❣✉②❡♥✲◆❣♦❝✱ P✐❡rr❡ ❆✉❣❡r✱ ❙✐❞② ▲②✱ ❉✐❞✐❡r ❏♦✉❢❢r❡ ✭✷✵✶✻✮ ❈❛♥ ❋✐s❤✐♥❣ Pr❡ss✉r❡ ■♥✈❡rt t❤❡ ❖✉t❝♦♠❡ ♦❢ ■♥t❡rs♣❡❝✐❢✐❝ ❈♦♠♣❡t✐t✐♦♥❄ ❚❤❡ ❈❛s❡ ♦❢ t❤❡ ❚❤✐♦❢ ❛♥❞ ♦❢ t❤❡ ❖❝t♦♣✉s ❆❧♦♥❣ t❤❡ ❙❡♥❡❣❛❧❡s❡ ❈♦❛st✳ ❆❝t❛ ❇✐♦t❤❡♦r❡t✐❝❛✱ ✻✹✱ ♣♣✳ ✺✶✾✲✺✸✻✳ ✭❙❈■✮ ✸✳ ❚❤✉② ◆❣✉②❡♥✲P❤✉♦♥❣✱ ❖❛♥❤ ❚r❛♥✲❚❤✐✲❑✐♠✱ ❉♦❛♥❤ ◆❣✉②❡♥✲◆❣♦❝✱ ❊❢❢❡❝ts ♦❢ ❋❛st ❉✐s♣❡rs❛❧ ❛♥❞ ❙t❛❣❡✲❙tr✉❝t✉r❡❞ ♦♥ Pr❡❞❛t♦r✲Pr❡② ❉②♥❛♠✐❝s✿ ❆ ❈❛s❡ ❙t✉❞② ♦❢ ❇r♦✇♥ P❧❛♥t✲❍♦♣♣❡r ❊❝♦❧♦❣✐❝❛❧ ❙②st❡♠✳ ✭❛❝❝❡♣t❡❞ ❢♦r ♣✉❜❧✐❝❛t✐♦♥ ✐♥ ❱✐❡t♥❛♠ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❆♣♣❧✐❝❛t✐♦♥✮ ✹✳ ❚❤✉② ◆❣✉②❡♥✲P❤✉♦♥❣✱ ❉♦❛♥❤ ◆❣✉②❡♥✲◆❣♦❝✱ ❉✉♦♥❣ P❤❛♥✲❚❤✐✲❍❛✱ ❖♥ t❤❡ ●❡♥❡r❛t✐♥❣ ●r❛♣❤ ♦❢ ❛♥ ■♥❞✐✈✐❞✉❛❧✲❇❛s❡❞ Pr❡❞❛t♦r✲Pr❡② ▼♦❞❡❧✳ ✭s✉❜♠✐tt❡❞✮ ... ✭❜✐♦t✐❝✮✱ t❛ ❝â γ(R) = rR(1 − R/K) ợ r K tữỡ ự tố t trữ ự ổ trữớ ỗ tự (R) = r(S R) ỗ tự ➠♥ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ s✐♥❤ ❤å❝ ✭❛❜✐♦t✐❝✮✱ ✈ỵ✐ r ❧➔ tố ỗ t S ỗ t tữỡ tỹ ữ ự ổ tr÷í♥❣✳ di ❧➔ t❤❛♠... ❦❤↔♦ s→t✳ ❈❤ó♥❣ tỉ✐ s♦ s tở t ỗ t s r tứ ❝→❝ ♠ỉ ❤➻♥❤ ❝→ t❤➸ ❝❤♦ ❤➺ ❝↕♥❤ tr❛♥❤ t❤ó ỗ ợ ởt số ổ tố ự t ❦❤→❝✳ ❈❤ó♥❣ tỉ✐ ❝ơ♥❣ t❤↔♦ ❧✉➟♥ ✈➲ ❝→❝ t➼♥❤ ❝❤➜t ♥➔② tø q✉❛♥ ✤✐➸♠ s✐♥❤ ❤å❝✳ ✸✳✷ ▼æ ❤➻♥❤ ❝↕♥❤... ✈➔ ①❛♥❤ ❧→ tữỡ ự ợ tú ỗ ọ ổ ỗ t s ổ t tr túỗ ổ ỗ t ự t ứ õ ỗ t t t t ởt số tố ự t tr t ợ ữ trt ỗ t ữủ ❏❡❛♥✲▲♦✉♣ ●✉✐❧❧❛✉♠❡ ✈➔ ▼❛tt❤✐❡✉ ▲❛t❛♣② tê♥❣ ❦➳t ✈➔♦ ♥➠♠ ✷✵✵✹✳ ◆❤ú♥❣