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VIETNAM NATIONAL UNIVERSITY, HANOI VNU UNIVERSITY OF SCIENCE Tran Dinh Tuong ASYMPTOTIC BEHAVIOR OF POPULATION MODELS IN ECOSYSTEM WITH RANDOM ENVIRONMENT THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS HANOI – 2020 VIETNAM NATIONAL UNIVERSITY, HANOI VNU UNIVERSITY OF SCIENCE Tran Dinh Tuong ASYMPTOTIC BEHAVIOR OF POPULATION MODELS IN ECOSYSTEM WITH RANDOM ENVIRONMENT Speciality: Differential and Integral Equations Speciality Code: 9460101.03 THESIS FOR THE DEGREE OF DOCTOR OF PHYLOSOPHY IN MATHEMATICS Supervisors: PROF DR NGUYEN HUU DU ASSOC PROF DR NGUYEN THANH DIEU HANOI – 2020 ĐẠI HỌC QUỐC GIA HÀ NỘI TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN Trần Đình Tướng DÁNG ĐIỆU TIỆM CẬN CỦA MỘT SỐ MƠ HÌNH QUẦN THỂ TRONG HỆ SINH THÁI VỚI MƠI TRƯỜNG NGẪU NHIÊN Chun ngành: Phương trình Vi phân Tích phân Mã số: 9460101.03 LUẬN ÁN TIẾN SĨ TOÁN HỌC Người hướng dẫn khoa học: GS TS NGUYỄN HỮU DƯ PGS TS NGUYỄN THANH DIỆU HÀ NỘI – 2020 Contents P❛❣❡ ❆❜str❛❝t ✐✈ Tóm tắt v ▲✐st ♦❢ ❋✐❣✉r❡s ✈✐ ▲✐st ♦❢ ❚❛❜❧❡s ✐① ▲✐st ♦❢ ◆♦t❛t✐♦♥s ① ■♥tr♦❞✉❝t✐♦♥ ✶ ❈❤❛♣t❡r ✶ ✶✷ Pr❡❧✐♠✐♥❛r✐❡s ✶✳✶ ❙t♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✶✳✶✳ ▼❛rt✐♥❣❛❧❡ ♣r♦❝❡ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✶✳✷✳ ▼❛r❦♦✈ ♣r♦❝❡ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✶✳✸✳ ▲é✈② ♣r♦❝❡ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✷ ❙t♦❝❤❛st✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✭❙❉❊s✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✷✳✶✳ ❙❉❊s ✇✐t❤ ▼❛r❦♦✈✐❛♥ ❙✇✐t❝❤✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✷✳✷✳ ❙❉❊s ✇✐t❤ ❥✉♠♣s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✶✳✸ Pr❡❧✐♠✐♥❛r✐❡s ❢♦r st♦❝❤❛st✐❝ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧s ✐♥ ❡❝♦s②st❡♠ ✳ ✳ ✳ ✳ ✸✶ ❈❤❛♣t❡r ✷ ▲♦♥❣✲t❡r♠ ❜❡❤❛✈✐♦r ♦❢ st♦❝❤❛st✐❝ ♣r❡❞❛t♦r✲♣r❡② s②st❡♠s ✸✹ ✷✳✶ ❉②♥❛♠✐❝ ❜❡❤❛✈✐♦r ♦❢ ❛ st♦❝❤❛st✐❝ ♣r❡❞❛t♦r✲♣r❡② s②st❡♠ ✉♥❞❡r r❡❣✐♠❡ s✇✐t❝❤✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✐ ✷✳✶✳✶✳ ■♥tr♦❞✉❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✶✳✷✳ ❙✉✣❝✐❡♥t ❛♥❞ ❛❧♠♦st ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥ ❢♦r ♣❡r♠❛♥❡♥❝❡ ✳ ✳ ✳ ✸✻ ✷✳✶✳✸✳ ❉✐s❝✉ss✐♦♥ ❛♥❞ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✷✳✷ ❖♥ t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ ❛ st♦❝❤❛st✐❝ ♣r❡❞❛t♦r✲♣r❡② ♠♦❞❡❧ ✇✐t❤ ■✈❧❡✈✬s ❢✉♥❝t✐♦♥❛❧ r❡s♣♦♥s❡ ❛♥❞ ❥✉♠♣s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✷✳✷✳✶✳ ■♥tr♦❞✉❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✷✳✷✳✷✳ ■♥tr♦❞✉❝t♦r② r❡s✉❧ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✷✳✷✳✸✳ ❆❧♠♦st ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ ❢♦r ❡①t✐♥❝t✐♦♥ ❛♥❞ ♣❡r♠❛♥❡♥❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✷✳✷✳✹✳ ❉✐s❝✉ss✐♦♥ ❛♥❞ ♥✉♠❡r✐❝❛❧ ❡①❛♠♣❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ❈❤❛♣t❡r ✸ ❊①t✐♥❝t✐♦♥ ❛♥❞ ♣❡r♠❛♥❡♥❝❡ ✐♥ ❛ st♦❝❤❛st✐❝ ❙■❘❙ ♠♦❞❡❧ ✐♥ r❡❣✐♠❡ s✇✐t❝❤✐♥❣ ✇✐t❤ ❣❡♥❡r❛❧ ✐♥❝✐❞❡♥❝❡ r❛t❡ ✼✶ ✸✳✶ ■♥tr♦❞✉❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✸✳✷ ❙✉✣❝✐❡♥t ❛♥❞ ❛❧♠♦st ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ❢♦r ♣❡r♠❛♥❡♥❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✸✳✸ ❉✐s❝✉ss✐♦♥ ❛♥❞ ♥✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ❈♦♥❝❧✉s✐♦♥ ✽✹ ❚❤❡ ❛✉t❤♦r✬s ♣✉❜❧✐❝❛t✐♦♥s r❡❧❛t❡❞ t♦ t❤❡ t❤❡s✐s ✽✺ ❆♣♣❡♥❞✐❝❡s ✽✻ ❇✐❜❧✐♦❣r❛♣❤② ✾✹ Acknowledgments The completion of this thesis could not have been possible without the guidance, assistance, and participation of so many people whose names may not all be enumerated Their contributions are greatly appreciated and gratefully acknowledged First and foremost, no words can express fully my gratitude and appreciation to my primary supervisor, Professor Nguyen Huu Du for his tireless support and endless guidance, and infusing spirit into my research I also wish to express the deepest thanks to the thesis co-supervisor, Associate Professor Nguyen Thanh Dieu for his kind comments, valued suggestions, and sharing great ideas during this course I am particularly grateful to Dr Nguyen Hai Dang for many valuable discussions and his great support during my work I own my thanks to Dr Tran Quan Ky for his contribution to our joint work [Pub 2] I would like to thank all the teachers, staff members and the management of the Faculty of Mathematics Mechanics and Information Technology at VNU University of Science, Vietnam National University, Hanoi as well as VIASM for their great support, lessons as well as opportunities for completion of the research I wish to thank my friends, who always trust, encourage and support me over the years Lastly, I would like to share this moment with my family I am indebted to my parents, my wife, Cherry, for their endless care, love and patience Hanoi, February 2020 PhD candidate Tran Dinh Tuong iii Abstract Abstract In this thesis, we consider long-term behavior of a class of formulated population models using stochastic differential equations to describe predator-prey relationships and explore the spread of infectious diseases For predator-prey systems, we study two models: one with both color and white noise and one with Ivlev’s functional response perturbed simultaneously by the white noise and Lévy noise For the study in the epidemic model, this thesis concerns a stochastic SIRS model, perturbed by both the white noise and the color noise, with a general incidence rate We propose new approaches to provide thresholds which indicate whether the systems are eventually extinct or permanent This allows us to derive not only sufficient conditions but also almost necessary conditions for permanence (as well as ergodicity) based on sign of such thresholds Furthermore, conditions for the existence of stationary distributions and for the validity of the strong law of large numbers are established in some particular cases Keywords Lotka-Volterra equation, predator-prey system, asymptotic behavior, ergodicity, regime switching diffusion process, stationary distribution, Ivlev’s functional response, extinction, permanence, jump diffusion process, SIRS model, epidemic models iv Tóm tắt Tóm tắt Trong luận án chúng tơi nghiên cứu dáng điệu tiệm cận lớp mơ hình sinh thái, mơ tả phương trình vi phân ngẫu nhiên, để mô tả tương tác thú mồi nghiên cứu lan truyền dịch bệnh Đối với hệ thú-mồi, nghiên cứu hai mơ hình: mơ hình thứ với nhiễu trắng nhiễu màu, mơ hình thứ hai có đáp ứng chức dạng Ivlev bị chịu đồng thời nhiễu trắng nhiễu Lévy Đối với mơ hình dịch bệnh, luận án đề cập đến mơ hình tái nhiễm SIRS bị chịu nhiễu trắng nhiễu màu với hàm truyền bệnh tổng quát Chúng đề xuất phương pháp tiếp cận để xây dựng giá trị ngưỡng nhằm hệ đến lúc tuyệt chủng tồn bền vững Dựa vào dấu giá trị ngưỡng, thu điều kiện đủ mà gần với điều kiện cần cho tồn bền vững tính ergodic hệ Hơn nữa, điều kiện cho tồn phân phối dừng điều kiện cho luật số lớn có hiệu lực thiết lập trường hợp cụ thể Từ khóa Mơ hình Lotka-Volterra, mơ hình predator-prey, dáng điệu tiệm cận, tính ergodic, hệ khuếch tán có bước chuyển Markov, phân phối dừng, đáp ứng chức dạng Ivlev, tuyệt chủng, tồn bền vững, q trình khuếch tán có bước nhảy, mơ hình SIRS, mơ hình dịch bệnh v Declaration This work has been completed at VNU University of Science, Vietnam National University, Hanoi under the supervision of Prof Dr Nguyen Huu Du and Assoc Prof Nguyen Thanh Dieu I declare hereby that the results in this thesis, which are allowed by my coauthors to be presented in this thesis, are new and they have never been used in any other theses Author Tran Dinh Tuong vi List of Figures ✷✳✶ ❚r❛❥❡❝t♦r✐❡s ♦❢ y(t) ✐♥ t❤❡ st❛t❡ ✶ ✭❜❧✉❡ ❧✐♥❡✮ ❛♥❞ ✐♥ t❤❡ st❛t❡ ✷ ✭r❡❞ ❧✐♥❡✮ ✐♥ ❊①✳ ✷✳✶✳✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✷✳✷ ❆ s✇✐t❝❤✐♥❣ tr❛❥❡❝t♦r② y(t) ✐♥ ❊①✳ ✷✳✶✳✻ ♦♥ t❤❡ ❧❡❢t ❛♥❞ ❆ s✇✐t❝❤✐♥❣ tr❛❥❡❝t♦r② y(t) ✐♥ ❊①✳ ✷✳✶✳✼ ♦♥ t❤❡ r✐❣❤t✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✷✳✸ ❚r❛❥❡❝t♦r✐❡s ♦❢ y(t) ✐♥ t❤❡ ✜rst st❛t❡ ✭❜❧✉❡ ❧✐♥❡✮ ❛♥❞ t❤❡ s❡❝♦♥❞ st❛t❡ ✭r❡❞ ❧✐♥❡✮ r❡s♣❡❝t✐✈❡❧② ✐♥ ❊①✳ ✷✳✶✳✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✷✳✹ P❤❛s❡ ♣✐❝t✉r❡ ❛♥❞ ❡♠♣✐r✐❝❛❧ ❞❡♥s✐t② ♦❢ x(t), y(t) ✐♥ ❊①✳ ✷✳✶✳✼ ✐♥ ✷❉ ❛♥❞ ✸❉ s❡tt✐♥❣s r❡s♣❡❝t✐✈❡❧②✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✷✳✺ P❤❛s❡ ♣✐❝t✉r❡ ♦❢ (x(t), y(t)) ❛♥❞ ❡♠♣✐r✐❝❛❧ ❞❡♥s✐t② ♦❢ ✐♥✈❛r✐❛♥t ♠❡❛✲ s✉r❡ s❡tt✐♥❣s r❡s♣❡❝t✐✈❡❧② ✭✇✐t❤ λ1 = 1✮✳ ❉✐✛❡r❡♥t ❝♦❧♦rs r❡♣r❡s❡♥t ❞✐✛❡r❡♥t s✐③❡s ♦❢ t❤❡ ❞❡♥s✐t②✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✷✳✻ ❚r❛❥❡❝t♦r✐❡s ♦❢ x(t) ♦♥ t❤❡ ❧❡❢t ❛♥❞ ♦❢ y(t) ♦♥ t❤❡ r✐❣❤t ✇✐t❤ ✐♥t❡♥s✐t② λ1 = 6.667✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✷✳✼ ❚r❛❥❡❝t♦r✐❡s ♦❢ x(t) ♦♥ t❤❡ ❧❡❢t ❛♥❞ ♦❢ y(t) ♦♥ t❤❡ r✐❣❤t ✇✐t❤ ✐♥t❡♥s✐t② λ1 = 7.6923✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✸✳✶ ❙❛♠♣❧❡ ♣❛t❤s ♦❢ I(t) ✭✐♥ ❜❧✉❡ ♦♥ t❤❡ ❧❡❢t✮✱ S(t) ✭✐♥ ❜❧✉❡ ♦♥ t❤❡ r✐❣❤t✮✱ ❛♥❞ ξt ✭✐♥ r❡❞✮ ✐♥ ❊①✳ ✸✳✸✳✺✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ✸✳✷ ❙❛♠♣❧❡ ♣❛t❤s ♦❢ I(t) ✭✐♥ ❜❧✉❡ ♦♥ t❤❡ ❧❡❢t✮ ❛♥❞ S(t) ✭✐♥ ❜❧✉❡ ♦♥ t❤❡ r✐❣❤t✮ ❛♥❞ ξt ✭✐♥ r❡❞✮ ✐♥ ❊①✳ ✸✳✸✳✻✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸ ✈✐✐ Appendices This section is devoted to the proofs of some technical results in Chapter (Theorem A.1, Theorem A.2) and Chapter (Lemma A.3) of this thesis The following theorem provides a details of proof in [53] In that paper, the author proposed the model n aij (ξt )xj (t) dt + αi (ξt )xi (t)dB1 (t) dxi (t) = xi (t) ri (ξt ) + j=1 n σij (ξt )xi (t)xj (t)dB2 (t), + i = 1, , n (1) j=1 For any constant sequence {cij (k)}, (1 cu = max i,j n,k∈S n, k ∈ S), define i, j cij (k), cl = i,j n,k∈S cij (k) cu (k) = max cij (k), cl = cij (k) i,j n i,j n Theorem A.1 ([53, Theorem 7]) If for each u ∈ S, σii (u) > for σij (u) i n whilst for i = j, the solution x(t) of (1) obeys lim sup t→+∞ Proof Define W (x) = ln U (x) = ln ln |x(t)| ln t n i=1 xi 87 1, a.s By using the generalized Itô’s formula, we have n dW (x) = U (x) − n xi ri (ξ) + i=1 i=1 n 2U (x)  n dt j=1   + U (x) aij (ξ)xj αi (ξ)xi n n σij (ξ)xi xj + αi (ξ)xi dB1 (t) + U (x) n dt  i=1 j=1 i=1   n σij (ξ)xi xj dB2 (t) i=1 j=1 Applying the generalized Itô’s formula again, we obtain n xi + U (x) d(exp(t)W (x)) = exp(t) ln i=1  exp(t)  − 2U (x)  exp(t) + U (x) n i=1 n n xi ri (ξ) + i=1 αi (ξ)xi dt j=1 n aij (ξ)xj n n σij (ξ)xi xj + exp(t) αi (ξ)xi dB1 (t) + U (x) n dt  i=1 j=1 i=1   n σij (ξ)xi xj dB2 (t) i=1 j=1 Thus, we have already shown that n n xi (t) − ln exp(t) ln i=1 i=1 t = xi (0) n xi (s) exp(s) ln i=1 + U (x(s)) t − n n n αi (ξs )xi (s) i=1 n + σij (ξs )xi (s)xj (s)   ds + N (t),  i=1 j=1 where t N (t) = exp(t) U (x(s)) t + ds j=1  exp(s)  2U (x(s))  n aij (ξs )xj (s) xi (s) ri (ξs ) + i=1 n αi (ξs )xi (s)dB1 (s) i=1 exp(s) U (x(s)) n n σij (ξs )xi (s)xj (s)dB2 (s) i=1 j=1 88 (2) The quadratic form of N (t) is t N (t), N (t) =  exp(2s)  U (x(s))  n n αi (ξs )xi (s) i=1 n σij (ξs )xi (s)xj (s) +   ds  i=1 j=1 By the exponential martingale inequality, we obtain that sup [N (t) − 0.5ε exp(−τ k) N (t), N (t) ] > ε−1 ρ exp(τ k) ln k P k −ρ , t τk where < ε < 1, ρ > and τ > By virtue of the Borel-Cantelli lemma, for almost all ω ∈ Ω, there exists k0 (ω) such that for every k k0 (ω), 0.5ε exp(−τ k) N (t), N (t) + ε−1 ρ exp(τ k) ln k, N (t) t τ k In other words, t N (t) exp(2s) U (x(s)) 0.5ε exp(−τ k) t + 0.5ε exp(−τ k) n αi (ξs )xi (s) ds i=1 exp(2s) U (x(s)) n n σij (ξs )xi (s)xj (s) ds i=1 j=1 + ε−1 ρ exp(τ k) ln k for t τ k Substituting this inequality into Equation (2) gives n n xi (t) − ln exp(t) ln = xi (0) i=1 i=1 t n xi (s) exp(s) ln i=1 + U (x(s)) t − t − n n xi (s) ri (ξs ) + i=1 exp(s) 2U (x(s)) exp(2s) U (x(s)) aij (ξs )xj (s) ds j=1 n αi (ξs )xi (s) [1 − ε exp(s − τ k)]ds i=1 n n σij (ξs )xi (s)xj (s) [1 − ε exp(s − τ k)]ds i=1 j=1 + ε−1 ρ exp(ξk) ln k (3) 89 Since s τ k, then t − exp(s) 2U (x(s)) n [1 − ε exp(s − τ k)]ds < αi (ξs )xi (s) i=1 At the same time U (x) n n n u |r| + |a| aij (ξ)xj xi ri (ξ) + j=1 i=1 u xi i=1 Moreover, there exists a positive constant ν such that t n exp(2s) U (x(s)) n [1 − ε exp(s − τ k)]ds σij (ξs )xi (s)xj (s) i=1 j=1 t ν n exp(s) xi (s) [1 − ε exp(s − τ k)]ds i=1 Then it follows from (3) that n n xi (t) − ln exp(t) ln xi (0) i=1 n i=1 t xi (s) + |r| + |a| exp(s) ln − ν n u i=1 n xi (s) i=1  [1 − ε exp(s − τ k)] ds xi (s) i=1 + ε−1 ρ exp(τ k) ln k K[exp(t) − 1] + ε−1 ρ exp(τ k) ln k, where K is a positive number If τ (k − 1) n t τ k and k k0 (ω), we have n xi (t)/ ln t ln u i=1 xi (0)/ ln t exp(−t) ln i=1 + K[1 − exp(−t)]/ ln t + ε−1 ρ exp(−τ (k − 1)) exp(τ k) ln k/ ln t a.s That is to say n ln i=1 xi (t) ln |x(t)| lim supt→+∞ lim supt→+∞ ε−1 ρ exp(τ ) ln t ln t Letting ε → 1, ρ → and τ → leads to the desired assertion The following theorem provides a details of proof in [63] which is applied in Part i) of Theorem 2.1.2 In that paper, the authors proposed the model dx(t) = diag(x1 (t), , xn (t))[(b + Ax(t)) dt + σx(t)dB(t)], Here σ = (σij )n×n satisfying σii > , if 90 i n; σij 0, if i = j Then (4) Theorem A.2 ([63, Theorem 2]) Let the system parameters b ∈ Rn and A ∈ Rn×n be given Then, for any θ ∈ (0, 1), there exists a positive constant Kθ such that, for any initial value x0 ∈ Rn,o + , the solution of Equation (4) has the property t lim sup E t→∞ t n x2+θ (s)ds i Kθ a.s i=1 n θ i=1 xi n,o Proof Define a C -function V : Rn,o + → R+ by V (x) = According to the Itô’s formula,  n n θxi bi + dV (x(t)) =  aij xj i=1 + j=1 n n n θ(θ − 1)xθi xj σij   dt j=1 i=1 n θxθi + σij xj dB(t) i=1 j=1 Moreover, it is easy to show that and n θ)xθi θ(1 − n θ(1 − θ)x2+θ σii2 i σij i=1 |aij |xi xj i=1 j=1 i=1 j=1 n n θxi |bi | + aij xj θxi bi + i=1 n n n n j=1 i=1 As a result, we obtain n n dV (x(t)) n |bi |xi + θ i=1 i=1 n n σii2 x2+θ dt i i=1 n xθi +θ θ(1 − θ) |aij |xi xj − j=1 i=1 σij xj dB(t) (5) j=1 Furthermore, by taking into consideration the fact that the polynomial n n |bi |xi + θ i=1 i=1 n θ(1 − θ) |aij |xi xj − j=1 n σii2 x2+θ i i=1 has an upper positive bound, say Kθ , inequality (5) yields θ(1 − θ) V (x(t)) + where M (t) = θ t n θ i=1 xi t n t σii2 x2+θ i i=1 V (x(0)) + Kθ ds + M (t), (6) n j=1 σij xj dB(s) is a real-valued continuous local mar- tingale vanishing at t = Taking expectations on both sides of (6), this completes the proof 91 The following Lemma is investigated by Benaăm [6] The Lemma is a key technique in our finding in Chapter of the thesis Another version of the Lemma can be found in [17] Lemma A.3 Let Y be a random variable, suppose E exp(Y ) + E exp(−Y ) K1 Then the log-Laplace transform u(θ) = ln E exp(θY ) is twice differentiable on [0, 0.5] and du dθ (0) d2 u dθ2 (θ) = EY, K2 , θ ∈ [0, 0.5] for some K2 > depending only on K1 Thus, it follows from Taylor’s expansion that EY θ + K2 θ2 , θ ∈ [0, 0.5] u(θ) Proof It is easy to show that there exists some K > such that |y|k exp(θy) K(exp(y) + exp(−y)), k = 1, for θ ∈ 0, 12 , y ∈ R For any y ∈ R, let ξ(y) be a number lying between y and ey − such that exp(ξ(y)) = Pick θ ∈ 0, 21 and let h ∈ R such that θ + h 21 y Then exp((θ + h)Y ) − exp(θY ) lim = Y exp(θY ) a.s., and h→0 h exp((θ + h)Y ) − exp(θY ) = |Y | exp(θY + ξ(hY )) 2K3 [exp(Y ) + exp(−Y )] h By the Lebesgue dominated convergence theorem, dE exp(θY ) exp((θ + h)Y ) − exp(θY ) = lim E = E[Y exp(θY )] h→0 dθ h Similarly, d2 E exp(θY ) = E[Y exp(θY )] dθ 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