Luận án tiến sĩ dáng điệu tiệm cận của một số mô hình quần thể trong hệ sinh thá

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. NGUYEN HUU DU ASSOC. 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 MÔ HÌNH QUẦN THỂ TRONG HỆ SINH THÁI VỚI MÔI TRƯỜNG NGẪU NHIÊN Chuyên ngành: Phương trình Vi phân và Tích phân Mã số: 9460101.03 LUẬN ÁN TIẾN SĨ TOÁN HỌC Người hướng dẫn khoa học: GS. NGUYỄN HỮU DƯ PGS.

NGUYỄN THANH DIỆU HÀ NỘI – 2020 Contents Page Abstract iv Tóm tắt v List of Figures vi List of Tables ix List of Notations x Introduction 1 Chapter 1 Preliminaries 12 1.2 Stochastic dierential equations (SDEs). SDEs with Markovian Switching. SDEs with jumps .3 Preliminaries for stochastic mathematical models in ecosystem. 31 Chapter 2 Long-term behavior of stochastic predator-prey systems 34 2.1 Dynamic behavior of a stochastic predator-prey system under regime switching.

Sucient and almost necessary condition for permanence. Discussion and numerical solutions .2 On the asymptotic behavior of a stochastic predator-prey model with Ivlev's functional response and jumps. Almost necessary and sucient condition for extinction and permanence. Discussion and numerical example.

67 Chapter 3 Extinction and permanence in a stochastic SIRS model in regime switching with general incidence rate 71 3.2 Sucient and almost necessary conditions for permanence .3 Discussion and numerical experiments. 79 Conclusion 84 The author's publications related to the thesis 85 Appendices 86 Bibliography 94 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 enu- merated. 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 Pro- fessor 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. 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 sys- tems, 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, per- turbed 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 ergodic- ity) 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. Lotka-Volterra equation, predator-prey system, asymptotic behavior, ergodicity, regime switching diffusion process, stationary distribution, Ivlev’s func- tional response, extinction, permanence, jump diffusion process, SIRS model, epi- demic models. iv Tóm tắt Tóm tắt. Trong luận án này chúng tôi nghiên cứu dáng điệu tiệm cận của lớp các mô hình sinh thái, được mô tả bằng các phương trình vi phân ngẫu nhiên, để mô tả sự tương tác giữa thú và mồi cũng như nghiên cứu sự lan truyền của dịch bệnh.

Đối với các hệ thú-mồi, chúng tôi nghiên cứu hai mô hình: mô hình thứ nhất với nhiễu trắng và nhiễu màu, mô hình thứ hai có đáp ứng chức năng dạng Ivlev bị chịu đồng thời cả nhiễu trắng và nhiễu Lévy. Đối với mô hình dịch bệnh, luận án đề cập đến mô hình tái nhiễm SIRS bị chịu cả nhiễu trắng và nhiễu màu với hàm truyền bệnh tổng quát. Chúng tôi đề xuất các phương pháp tiếp cận mới để xây dựng các giá trị ngưỡng nhằm chỉ ra hệ đến một lúc nào đó sẽ tuyệt chủng hoặc sẽ tồn tại bền vững. Dựa vào dấu của các giá trị ngưỡng, chúng tôi không những chỉ thu được điều kiện đủ mà còn rất gần với điều kiện cần cho sự tồn tại bền vững cũng như tính ergodic của hệ.

Hơn nữa, các điều kiện cho sự tồn tại của các phân phối dừng cũng như các điều kiện cho luật số lớn có hiệu lực được thiết lập trong các trường hợp cụ thể. Mô hình Lotka-Volterra, mô hình predator-prey, dáng điệu tiệm cận, tính ergodic, hệ khuếch tán có bước chuyển Markov, phân phối dừng, đáp ứng chức năng dạng Ivlev, sự tuyệt chủng, sự tồn tại bền vững, quá trình khuếch tán có bước nhảy, mô hình SIRS, các mô hình dịch bệnh. v Declaration This work has been completed at VNU University of Science, Vietnam National University, Hanoi under the supervision of Prof. Nguyen Huu Du and Assoc.

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 2.1 Trajectories of y(t) in the state 1 (blue line) and in the state 2 (red line) in Ex.2 A switching trajectory y(t) in Ex.6 on the left and A switching trajectory y(t) in Ex.7 on the right.3 Trajectories of y(t) in the rst state (blue line) and the second state (red line) respectively in Ex.4 Phase picture and empirical density of x(t), y(t) in Ex.7 in 2D
and 3D settings respectively.5 Phase picture of (x(t), y(t)) and empirical density of invariant mea- sure settings respectively (with λ1 = 1).

Dierent colors represent dierent sizes of the density.6 Trajectories of x(t) on the left and of y(t) on the right with intensity λ1 = 6.7 Trajectories of x(t) on the left and of y(t) on the right with intensity λ1 = 7.1 Sample paths of I(t) (in blue on the left), S(t) (in blue on the right), and ξt (in red) in Ex.2 Sample paths of I(t) (in blue on the left) and S(t) (in blue on the right) and ξt (in red) in Ex.3 Empirical density of (S(t), I(t)) with respect to the rst state p(i; s; 1) (on the left) and in the second state p(i; s; 2) (on the right) of an invariant probability measure of Ex.4 Empirical density of (S(t), I(t)) with respect to the rst state p(i; s; 1) (on the left) and in the second state p(i; s; 2) (on the right) of an invariant probability measure of Ex. 84 viii List of Tables 2.1 Values of the coecients in Example 2.2 Values of the coecients in Example 2.1 Values of the coecients in Example 3.2 Values of the coecients in Example 3. 83 ix List of Notations i.d independent, identically distributed. Ac complement of set A.

1A (x) the indicator function of a Borel set A. N the set of natural numbers. R the real line. Rn the n-dimensional Euclidean space.

B(S) Borel σ-algebra of S ⊂ Rn. Bh the open ball centered at 0 with radius h > 0. (Ω, F, P) the probability space. P(A) the probability of event A.

EX expectation of random variable X. Px the probability with the initial condition. Ex the expectation corresponding with the initial condition. a∧b the minimum of a and b.

a∨b the maximum of a and b. ξt , t > 0 right continuous Markov chain., N }, the finite state space of a Markov chain. Γ (qij )N ×N the generator of the Markov chain. trace A the trace of a square matrix A = (Aij )n×n , P i.

trace A = 16i6n Aii A> the transpose of a vector or matrix A. |x| the Euclidean norm of a vector x. Other notations will be explained where they first appear. x Introduction Mathematical biology (or biomathematics) is a part of biology which utilizes hypo- thetical examination, mathematical models and abstractions of the living organisms to investigate the principles that govern the structure, development and behavior of the systems, as opposed to experimental biology which manages the conduction of investigations to prove and validate the scientific theories which stresses the math- ematical side while theoretical biology stresses the biological side.

Because of the complexity of biological systems, mathematical biology examines a variety areas of specialized research: computer models and automata theory, mathematical bio- physics, molecular set theory, etc. and it tends to be helpful in both theoretical and practical research. This thesis belongs to the field of mathematical biology. Although there are many ways to think about the theory of mathematical biology, it is interesting to look at this field as a theory of ordinary differential equations (ODEs for short).

By modeling method, ODEs allow us to test and generate hypotheses, and help us describe the evolution in times of population. That is the simplest way to study the development of a population viewed as a dynamical system. The investigation of population change by ODEs has a long term history. In his book [60] entitled “An essay on the principle of population”, Malthus observed that an increase in a nation’s food production improved the well-being of the populace, but the improvement was temporary because it led to population growth, which in turn restored the original per capita production level, and he proposed the simplest population of single species.

Observing that there must be adjustment to such ex- ponential growth, in the long run of course, Verhulst [81] introduced the logistic model. In order to study of the interrelationship between species and their envi- 1 ronment, in such areas as predator-prey and competition interactions, in [57] Lotka studied autocatalytic and the theory of competing species which was later followed by Volterra blue by proposing the simplest Lotka-Volterra model also known as the predator-prey model. After that, the most general growth model, namely Kol- mogorov type, was introduced by Kolmogorov in 1936 which was further developed by May, Rescigno and Richardson and Albrecht (see [76]). On the other hand, many epidemic models that describe the dynamics of the spread of infectious diseases have been analyzed mathematically and applied from the Kol- mogorov type.

In this field, the models focus on concepts and methods of mathemat- ical modeling of infectious diseases and can project how infectious diseases progress to show the likely outcome of an epidemic and help inform public health interven- tions.