Asymptotic behavior of population models in ecosystem with random environment

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 As a result, we obtain E[Y exp(θY )] du = dθ E exp(θY ) which implies du (0) = EY dθ and du = dθ2 E[Y exp(θY )]E exp(θY ) E[Y exp(Y )] [E exp(Y )] By Hăolders inequality we have E[Y exp(θY )]E exp(θY ) therefore d2 u dθ2 for all θ ∈ [0, 0.5] 92 E[Y exp(θY )] and Moreover, E[Y exp(θY )] K(E exp(Y ) + E exp(−Y )) E exp(θY ) exp(θEY ) K(E exp(Y ) + E exp(−Y )) := K2 , exp(−|EY |) which concludes the proof d2 u dθ2 93 Bibliography [1] Anderson R M., May R M (1978), “Regulation and stability of host-parasite population interactions I: Regulatory processes”, J Anim Ecol 47 (1), pp 219-267 [2] Applebaum D (2009), Lévy Processes and Stochastic Calculus, Second edition, Cambridge University Press [3] Arnold L., Horsthemke W., Stucki, J W (1979), “The influence of external real and white noise on the Lotka-Volterra model”, Biometrical J 21 (5), pp 451-471 [4] Asmussen S., Taksar M (1997), “Controlled diffusion models for optimal dividend pay-out”, Insurance Math Econom 20 (1), pp 1-15 [5] Bao J., Mao X., Yin G., Yuan C (2011), “Competitive Lotka-Volterra population dynamics with jumps”, Nonlinear Anal 74, pp 6601-6616 [6] Benaăm M., Lobry C (2016), Lotka-Volterra with randomly fluctuating environments or “How switching between beneficial environments can make survival harder” ”, Ann Appl Probab 26 (6), pp 3754-3785 [7] Brauer F (2017), “Mathematical epidemiology: past, present, and future”, Infectious Disease Modelling (2), pp 113-127 [8] Cai Y., Kang Y., Banerjee M., Wang W (2015), “A stochastic SIRS epidemic model with infectious force under intervention strategies”, J Differential Equations 259, pp 7463-7502 [9] Capasso V., Serio G (1978), “A generalization of Kermack-McKendrick deterministic epidemic model”, Math Biosci 42 (1–2), pp 43-61 94 [10] Chen G., Li T., Liu C (2014), “Lyapunov exponent and almost sure asymptotic stability of a stochastic SIRS model”, Publ Mat 58, pp 153-165 [11] Choulli T., Taksar M., Zhou X Y (2003), “A diffusion model for optimal dividend distribution for a company with constraints on risk control”, SIAM J Control Optim 41 (6), pp 1946-1979 [12] Cloez B., Hairer M (2015), “Exponential ergodicity for Markov processes with random switching”, Bernoulli 21, pp 505-536 [13] Dai M., Zhang Q., Zhu Q J (2010), “Trend following trading under a regime switching model”, SIAM J Financial Math (1), pp 780-810 [14] N H Dang (2014), “A note on sufficient conditions for asymptotic stability in distribution of stochastic differential equations with Markovian switching”, Nonlinear Anal 95, pp 625-631 [15] N H Dang, N H Du, T V Ton (2011), “Asymptotic behavior of predator-prey systems perturbed by white noise”, Acta Appl Math 115, pp 351-370 [16] N H Dang, N H Du, Yin G (2014), “Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise”, J Differential Equations 257, pp 2078-2101 [17] N H Dang, Yin G (2018), “Stability of regime switching diffusion systems with discrete states belonging to a countable set”, SIAM J Control Optim 56, pp 3893-3917 [18] N H Dang, Yin G., Zhu C (2017), “Certain properties related to well posedness of switching diffusions”, Stochastic Process Appl 127, pp 3135-3158 [19] N H Dang, Yin G (2017), “Coexistence and exclusion of stochastic competitive Lotka-Volterra models”, J Differential Equations 262 (3), pp 1192-1225 [20] N T Dieu, N H Dang, N H Du, Yin G (2016), “Classification of asymptotic behavior in a stochastic SIR model”, SIAM J Appl Dyn Syst 15 (2), pp 1062-1084 [21] N H Du, N H Dang (2011), “Dynamics of Kolmogorov systems of competitive type under the telegraph noise”, J Differential Equations 250, pp 386-409 95 [22] N H Du, N H Dang (2014), “Asymptotic behavior of Kolmogorov systems with predator-prey type in random environment”, Commun Pure Appl Anal 13(6), pp 2693-2712 [23] N H Du, N H Dang, Yin G (2016), “Conditions for permanence and ergodicity of certain stochastic predator-prey models”, J Appl Probab 53, pp 187-202 [24] N H Du, V H Sam (2006), “Dynamics of a stochastic Lotka-Volterra model perturbed by white noise”, J Math Anal Appl 324, pp 82-97 [25] Fubao X., Chao Z (2017), “On Feller and strong Feller properties and exponential ergodicity of regime switching jump diffusion processes with countable regimes”, SIAM J Control Optim 55 (3), pp 1789-1818 [26] Goodwin R M (1967), A Growth Cycle In: Essays in Economic Dynamics, Palgrave Macmillan, London [27] Gray A., Greenhalgh D., Hu L., Mao X , Pan J (2011), “A stochastic differential equation SIS epidemic model”, SIAM J Appl Math 71 (3), pp 876-902 [28] Greenhalgh D., Liang Y., Mao X (2016), “Modelling the effect of telegraph noise in the SIRS epidemic model using Markovian switching”, Phys A 462, pp 684-704 [29] Han Z., Zhao J (2013), “Stochastic SIRS model under regime switching”, Nonlinear Anal Real World Appl 14 (1), pp 352-364 [30] Hethcote H W (2000), “The mathematics of infectious diseases”, SIAM Rev 42 (4), pp 599-653 [31] Higham D J (2001), “An algorithmic introduction to numerical simulation of stochastic differential equations”, SIAM Rev 43, pp 525-546 [32] N T Hieu (2014), Mathematical population dynamics models in deterministic and stochastic environments, Université Pierre et Marie Curie-Paris VI, PhD thesis [33] Hening A., N H Dang (2018), “Coexistence and extinction for stochastic Kolmogorov systems”, Ann Appl Probab 28 (3), pp 1893-1942 96 [34] Holling C.S (1959), “The components of predation as revealed by a study of small-mammal predation of the European pine sawfly”, Canadian Entomologist 91, pp 293-320 [35] Ichihara K., Kunita H (1977), “A classification of the second order degenerate elliptic operators and its probabilistic characterization”, Z Wahrsch Verw Gebiete 30, pp 235-254 Corrections in 39 (1977), pp 81-84 [36] Ivlev V I (1955), Experimental Ecology of the Feeding of Fishes, Yale University Press, New Haven, Conn, USA [37] Ji C., Jiang D (2011), “Dynamics of a stochastic density dependent predatorprey system with Beddington-DeAngelis functional response”, J Math Anal Appl 381 (1), pp 441-453 [38] Karatzas I., Shreve S E (1991), Brownian Motion and Stochastic Calculus, Springer-Verlag, Berlin [39] Kermack W O., McKendrick A G (1927), “Contributions to the mathematical theory of epidemics”, (part I), Proc R Soc Lond Ser A 115, pp 700-721 [40] Kermack W O., McKendrick A G (1932), “Contributions to the mathematical theory of epidemics”, (part II), Proc Roy Sot Ser A 138, pp 55-83 [41] Khasminskii R A (2012), Stochastic Stability of Differential Equations, Springer-Verlag Berlin Heidelberg [42] Khasminskii R Z., Zhu C., Yin G (2007), “Stability of regime switching diffusions”, Stochastic Process Appl 117 (8), pp 1037-1051 [43] Kliemann W (1987), “Recurrence and invariant measures for degenerate diffusions”, Ann Probab 15 (2), pp 690-707 [44] Kooij R (1996), “A predator-prey model with Ivlev’s functional response”, J Math Anal Appl 198, pp 473-489 [45] Korobeinikov A., Wake G C (2002), “Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models”, Appl Math Lett 15 (8), pp 955-960 97 [46] Lahrouz A., Omari L., Kiouach D (2011), “Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model”, Nonlinear Anal Model Control 16 (1), pp 59-76 [47] Lahrouz A., Settati A (2013), “Asymptotic properties of switching diffusion epidemic model with varying population size”, Appl Math Comput 219 (24), pp 11134-11148 [48] Leah E K (2005), Mathematical Models In Biology, SIAMs Classics in Applied Mathematics, Random House, New York [49] Li D., Cui J., Song G (2015), “Permanence and extinction for a single-species system with jump diffusion”, J Math Anal Appl 430 (1), pp 438-464 [50] Li D., Cui J., Zhang Y (2015), “Permanence and extinction of non-autonomous Lotka-Volterra facultative systems with jump diffusion”, Discrete Contin Dyn Syst Ser B 20 (7), pp 2069-2088 [51] Li X., Mao X (2009), “Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation”, Discrete Contin Dyn Syst 24 (2), pp 523-545 [52] Lipster R (1980), “A strong law of large numbers for local martingales”, Stochastics, 3, pp 217-228 [53] Liu M (2014), “Dynamics of a stochastic Lotka-Volterra model with regime switching”, J Appl Math Comput 45, pp 327-349 [54] Liu M., Wang, K (2011), “Global stability of a nonlinear stochastic predatorprey system with Beddington-DeAngelis functional response”, Commun Nonlinear Sci Numer Simul 16, pp 1114-1121 [55] Liu W M., Levin S A., Iwasa Y (1986), “Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models”, J Math Biol 23, pp 187-204 [56] Liu Q., Jiang D., Shi N (2017), “Stochastic mutualism model with Lévy jumps”, Commun Nonlinear Sci Numer Simul 43, pp 78-90 98 [57] Lotka A J (1920), “Analytical note on certain rhythmic relations in organic systems”, Proc Natl Acad Sci USA (7), pp 410-415 [58] Lu Q (2009), “Stability of SIRS system with random perturbations”, Phys A 388, pp 3677-3686 [59] Luo Q., Mao X (2007), “Stochastic population dynamics under regime switching”, J Math Anal Appl 334, pp 69-84 [60] Malthus T R (1798), An Essay on the Principle of Population, 1a Ed, J Johnson in St Paul’s Churchyard, London [61] Mao X (1997), Stochastic Differential Equations and Applications, Horwood Publishing Chichester, New York [62] Mao X., Yuan C (2006), Stochastic Differential Equations with Markovian switching, Imperial College Press, London [63] Mao X., Sabanis S., Renshaw E (2003), “Asymptotic behaviour of the stochastic Lotka-Volterra model”, J Math Anal Appl 287, pp 141-156 [64] Meiss J D (2007), Differential dynamical systems, SIAM, Philadelphia, Pa, USA [65] Meyn S P., Tweedie R L (2009), Markov Chains and Stochastic Stability, Cambridge University Press, New York [66] Maruyama G., Tanaka H (1959), “Ergodic property of N-dimensional recurrent Markov processes”, Mem Fac Sci Kyushu Ser A 13, pp 157-172 [67] Martcheva M (2015), An Introduction to Mathematical Epidemiology, Springer Science + Business Media, New York [68] Murray J D (2002), Mathematical Biology I, Springer-Verlag, New York [69] Murray J D (2003), Mathematical Biology II, Springer-Verlag, New York [70] N’zi M., Tano J (2017), “Deterministic and stochastic stability of an SIRS epidemic model with a saturated incidence rate”, Random Oper Stoch Equ 25 (1), pp 11-26 99 [71] Øksendal B., Sulem A (2019), Applied Stochastic Control of Jump Diffusions, Third edition, Springer, Switzerland [72] Peng S., Zhu X (2006), “Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations”, Stochastic Process Appl 116, pp 370-380 [73] Pinsky M., Pinsky R (1993), “Transience recurrence and central limit theorem behavior for diffusions in random temporal environments”, Ann Probab 21, pp 433-452 [74] Rong S T (2005), Theory of Stochastic Differential Equations with Jumps and Applications, Springer, Berlin [75] Rudnicki R (2003), “Long-time behaviour of a stochastic prey-predator model”, Stochastic Process Appl 108, pp 93-107 [76] Sattar M A (1986), Perturbations and bifurcations in the three dimensional Kolmogorov model, Simon Fraser University, PhD thesis [77] Settati A., Lahrouz A., El Jarroudi M (2016), “Dynamics of hybrid switching diffusions SIRS model”, J Appl Math Comput 52 (1-2), pp 101-123 [78] Siakalli M (2009), Stability properties of stochastic differential equations driven by Lévy noise, University of Sheffield, PhD thesis [79] Sternberg S (2010), Dynamical Systems, Dover Publications (eTextbook) [80] Stettner L (1986), “On the existence and uniqueness of invariant measure for continuous time Markov processes”, LCDS Report, pp 86-16 [81] Verhulst P F (1838), “Notice sur la loi que la population poursuit dans son accroissement”, Correspondance mathématique et physique 10, pp 113-121 [82] Xi F (2009), “Asymptotic properties of jump diffusion processes with statedependent switching”, Stochastic Process Appl 119, pp 2198-2221 [83] Yang, Z., Mao, X., Yuan, C (2008), “Comparison theorem of one-dimensional stochastic hybrid systems”, Syst Control Lett 57, pp 56-63 100 [84] Yuan C., Mao X (2003), “Asymptotic stability in distribution of stochastic differential equations with Markovian switching”, Stochastic Process Appl 79, pp 45-67 [85] Zeng T., Teng Z., Li Z., Hu J (2018), “Stability in the mean of a stochastic three species food chain model with general Lévy jumps”, Chaos Solitons Fractals 106, pp 258-265 [86] Yin G., Zhu C (2009), Hybrid Switching Diffusions: Properties and Applications, Springer Science & Business Media, New York [87] Yousef A M., Salman S M (2016), “Backward bifurcation in a fractional-order SIRS epidemic model with a nonlinear incidence rate”, Int J Nonlinear Sci Numer Simul 17 (7-8), pp 401-412 [88] Zhu C (2011), “Optimal control of the risk process in a regime switching environment”, Automatica 47, pp 1570-1579 [89] Zhu C., Yin G (2007), “Asymptotic properties of hybrid diffusion systems”, SIAM J Control Optim 46 (4), pp 1155-1179 [90] Zhao Y., Jiang D (2014), “The threshold of a stochastic SIRS epidemic model with saturated incidence”, Appl Math Lett 34, pp 90-93 [91] Zhang X., Li W., Liu M., Wang K (2015), “Dynamics of a stochastic Holling II one-predator two-prey system with jumps”, Phys A 421, pp 571-582 [92] Zhang Q M., Jiang D Q., Zhao Y N., O’Regan D (2017), “Asymptotic behavior of a stochastic population model with Allee effect by Lévy jumps”, Nonlinear Anal Hybrid Syst 24, pp 1-12 101 ... 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... corresponding with the initial condition a∧b the minimum of a and b a∨b the maximum of a and b ξt , t 0, , xn 0} right continuous Markov chain S {1, 2, , N }, the finite state space of a Markov chain... early reference of epidemic models is made in a paper by Hamer [7] In his work, the probability of an infection in the next period of time was proportional to the number of infectious individuals