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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 I HC QUăC GIA H TR×˝NG I H¯C KHOA H¯C TÜ NHI N Trƒn D NG NáI nh Tữợng I UTI MC NCếAMáTSă M H NH QU N TH TRONG H SINH TH I VI MI TRìNG NG U NHI N Chuyản ng nh: Phữỡng trnh Vi phƠn v Tch phƠn M s: 9460101.03 LU N NTI NS TO NHC Ngữới hữợng dÔn khoa håc: GS TS NGUY N HÚU D× PGS TS NGUY N THANH DI U H N¸I 2020 Contents Abstract Tâm t›t List of Figures List of Tables List of Notations Introduction Chapter 1.1 Stochastic processes 1.1.1 1.1.2 1.1.3 LÈvy process Stochastic dierential equations (SDE 1.2 1.2.1 SDEs with Markovian Switch 1.2.2 1.3 Preliminaries for stochastic mathema Chapter Long-term behavior of stochastic predator-prey systems 2.1 Dynamic behavior of a stochastic pre switching i 2.1.1 Introduction 2.1.2 2.1.3 2.2 On the asymptotic behavior of a stoc Ivlev's functional response and jump 2.2.1 Introduction 2.2.2 Introductory results 2.2.3 Almost necessary and sucien 2.2.4 Discussion and numerical ex Chapter Extinction and permanence in a stochastic SIRS model in regime switching with general incidence rate 3.1 Introduction 3.2 Sucient and almost necessary cond 3.3 Discussion and numerical experime Conclusion The author's publications related to the thesis Appendices Bibliography 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 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 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 L†vy 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 Keywords 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 Tâm t›t Tâm t›t Trong lu“n ¡n n y chóng tỉi nghi¶n cøu d¡ng i»u ti»m c“n ca lợp cĂc mổ hnh sinh thĂi, ữổc mổ tÊ bng cĂc phữỡng trnh vi phƠn ngÔu nhiản, mổ tÊ sỹ tữỡng tĂc gia thú v cụng nhữ nghiản cứu sỹ lan truyãn ca dch bằnh i vợi cĂc hằ thú-mỗi, chúng tổi nghiản cứu hai mổ hnh: mổ hnh thứ nhĐt vợi nhiu trng v nhiu m u, mỉ h…nh thø hai câ ¡p øng chøc n«ng dng Ivlev b chu ỗng thới cÊ nhiu trng v nhi„u L†vy Łi vỵi mỉ h…nh dàch b»nh, lu“n ¡n • c“p ‚n mæ h…nh t¡i nhi„m SIRS bà chàu c£ nhi„u tr›ng v nhi„u m u vỵi h m truyãn bằnh tng quĂt Chúng tổi ã xuĐt cĂc phữỡng phĂp tip cn mợi xƠy dỹng cĂc giĂ tr ng÷ïng nh‹m ch¿ h» ‚n mºt lóc n o õ s tuyằt chng hoc s tỗn ti bãn vng Dỹa v o dĐu ca cĂc giĂ tr ngữùng, chúng tổi khổng nhng ch thu ữổc iãu kiằn m cặn rĐt gn vợi iãu kiằn cn cho sỹ tỗn ti bãn vng cụng nhữ tnh ergodic ca hằ Hỡn na, cĂc iãu kiằn cho sỹ tỗn ti ca cĂc phƠn phi dng cụng nhữ cĂc iãu kiằn cho lut s lợn cõ hiằu lỹc ữổc thit lp cĂc tr÷íng hỉp cư th” Tł khâa Mỉ h…nh Lotka-Volterra, mỉ h…nh predator-prey, d¡ng i»u ti»m c“n, t ‰nh ergodic, h» khuch tĂn cõ bữợc chuyn Markov, phƠn phi dng, Ăp ứng chức nông dng Ivlev, sỹ tuyằt chng, sỹ tỗn ti bãn vng, quĂ trnh khuch tĂn cõ bữợc nhÊy, mæ h…nh SIRS, c¡c mæ h…nh dàch b»nh 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 2.1 Trajectories of y(t) in the state (blue line) and in the st line) in Ex 2.1.6 2.2 A switching trajectory y(t) in Ex 2.1.6 on the left and A s trajectory y(t) in Ex 2.1.7 on the right 2.3 Trajectories of y(t) in the rst state (blue line) and the se (red line) respectively in Ex 2.1.7 and 3D settings respectively 2.4 Phase picture and empirical density of x(t); y(t) 2.5 Phase picture of (x(t); y(t)) and empirical density of inva sure settings respectively (with = 1) Dierent colors r dierent sizes of the density 2.6 Trajectories of x(t) on the left and of y(t) on the right wit 1=6:667 2.7 Trajectories of x(t) on the left and of y(t) on the right wit 1=7:6923 3.1 Sample paths of I(t) (in blue on the left), S(t) (in blue on and t (in red) in Ex 3.3.5 3.2 Sample paths of I(t) (in blue on the left) and S(t) (in blue right) and vii t (in red) in Ex 3.3.6 Since s k; then Zt exp(s) " n Xi 2U (x(s)) At the same time U(x) X n xi ri( ) + i=1 Moreover, there exists a positive constant such that Z Then it follows from (3) that exp(t) ln where K is a positive number If (k n Xi ln =1 That is to say Letting " ! 1; ! and ! 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); : : : ; x Here = ( ij)n n satisfying ii > , if i n; ij > 0, if i 6= j Then 90 n nn Theorem A.2 ([63, Theorem 2]) Let the system parameters b R and A R be given Then, for any (0; 1), there exists a positive constant K such that, for n; o any initial value x0 R + , the solution of Equation (4) has the property Proof Define a C -function V : Itæ’s formula, n n X + X i=1 xiijxjdB(t): j=1 Moreover, it is easy to show that n xi X i=1 and As a result, we obtain dV (x(t)) " (5) Furthermore, by taking into consideration the fact that the polynomial (6) has an upper positive bound, say K , inequality (5) yields where M(t) = R P tingale vanishing at t= the proof 91 The following Lemma is investigated by Bena m [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 ) K 1: Then the log-Laplace transform u( ) = ln E exp( Y ) is twice differentiable on [0; 0:5] and du d d u (0) = EY; d2 ( ) K2 ; [0; 0:5] for some K > depending only on K1 Thus, it follows from Taylor’s expansion that u( ) EY + K2 ; [0; 0:5] : Proof It is easy to show that there exists some K > such that k jyj exp( y) K(exp(y) + exp( y)); k = 1; 2: such for 0; that exp( (y)) = Then lim h!0 exp(( + h) h By the Lebesgue dominated convergence theorem, Similarly, As a result, we obtain which implies du d (0) = EY ineq By Holder’s therefore 92 Moreover, du d E[ K( 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GIA H TRìNG I H¯C KHOA H¯C TÜ NHI N Trƒn D NG N¸I nh Tữợng I UTI MC NCếAMáTSă M H NH QU N TH TRONG H SINH TH I V˛I M˘I TR×˝NG NG U NHI N Chuyản ng nh: Phữỡng trnh Vi phƠn v Tch phƠn M s: 9460101.03... diffusion process, SIRS model, epi-demic models iv Tâm t›t Tâm t›t Trong lu“n ¡n n y chóng tổi nghiản cứu dĂng iằu tiằm cn ca lợp cĂc mỉ h…nh sinh th¡i, ÷ỉc mỉ t£ b‹ng c¡c ph÷ìng trnh vi phƠn ngÔu nhiản,... simply rewrite as P (0; x; t; A) = P (t; x; A) A Markov process X is called a strong Markov process if the following strong Markov property is satisfied: for any bounded Borel measurable function