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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 z 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 z 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 z Contents Page Abstract iv Tóm tắt v List of Figures vi List of Tables ix List of Notations x Introduction Chapter 12 Preliminaries 1.1 Stochastic processes 12 1.1.1 Martingale process 13 1.1.2 Markov process 14 1.1.3 Lévy process 19 1.2 Stochastic dierential equations (SDEs) 21 1.2.1 SDEs with Markovian Switching 21 1.2.2 SDEs with jumps 26 1.3 Preliminaries for stochastic mathematical models in ecosystem 31 Chapter Long-term behavior of stochastic predator-prey systems 34 2.1 Dynamic behavior of a stochastic predator-prey system under regime switching 35 i z 2.1.1 Introduction 35 2.1.2 Su cient and almost necessary condition for permanence 36 2.1.3 Discussion and numerical solutions 51 2.2 On the asymptotic behavior of a stochastic predator-prey model with Ivlev's functional response and jumps 55 2.2.1 Introduction 55 2.2.2 Introductory results 57 2.2.3 Almost necessary and su cient condition for extinction and permanence 61 2.2.4 Discussion and numerical example 67 Chapter Extinction and permanence in a stochastic SIRS model in regime switching with general incidence rate 71 3.1 Introduction 71 3.2 Su cient and almost necessary conditions for permanence 73 3.3 Discussion and numerical experiments 79 Conclusion 84 The author's publications related to the thesis 85 Appendices 86 Bibliography 94 z 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 z 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 z Tóm tắt Tóm tắt Trong luận án 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, trình khuếch tán có bước nhảy, mơ hình SIRS, mơ hình dịch bệnh v z 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 z List of Figures 2.1 Trajectories of y(t) in the state (blue line) and in the state (red line) in Ex 2.1.6 53 2.2 A switching trajectory y(t) in Ex 2.1.6 on the left and A switching trajectory y(t) in Ex 2.1.7 on the right 53 2.3 Trajectories of y(t) in the rst state (blue line) and the second state (red line) respectively in Ex 2.1.7 54 2.4 Phase picture and empirical density of x(t), y(t) in Ex 2.1.7 in 2D and 3D settings respectively 55 2.5 Phase picture of (x(t), y(t)) and empirical density of invariant measure settings respectively (with λ1 = 1) Dierent colors represent dierent sizes of the density 68 2.6 Trajectories of x(t) on the left and of y(t) on the right with intensity λ1 = 6.667 69 2.7 Trajectories of x(t) on the left and of y(t) on the right with intensity λ1 = 7.6923 69 3.1 Sample paths of I(t) (in blue on the left), S(t) (in blue on the right), and ξt (in red) in Ex 3.3.5 82 3.2 Sample paths of I(t) (in blue on the left) and S(t) (in blue on the right) and ξt (in red) in Ex 3.3.6 83 vii z Using this recursively we obtain Eφ,i I −θ (nT ) q n y −θ + C(1 − q n ) for φ ∈ ∆, i ∈ S, n ∈ N 1−q (3.16) For any t ∈ [nT, nT + T ], by (3.14) Eφ,i I −θ (t) Eφ,i I −θ (nT )eH−θ (t−nT ) Eφ,i I −θ (nT )eH−θ T This estimate together with (3.16) leads to C(1 − q n ) −θ n −θ Eφ,i I (t) q y + exp(H−θ T ) for t ∈ [nT, nT + T ] 1−q (3.17) Letting n → ∞ we obtain lim sup Eφ,i I −θ (t) = t→∞ C exp(H−θ T ) a.s 1−q (3.18) On the other hand, Pφ,i (I(t) > δ) = − Pφ,i (I −θ (t) > δ −θ ) > − δ θ Eφ,i I −θ (t) This and (3.18) imply that lim inf Pφ,i (I(t) > δ) > − δ −θ lim sup Eφ,i I −θ (t) = − δ θ K a.s., t→∞ t→∞ C exp(H−θ T ) This implies (3.12) where δ satisfies δ θ K ε The 1−q proof is thus completed where K = 3.3 Discussion and numerical experiments To highlight the contributions of this chapter, we compare our results with some of the recent developments in the literature For instance, authors in [77] considered the model dS(t) = µ(ξt ) − β(ξt )S(t)I(t) − µ(ξt )S(t) + γ(ξt )R(t) −σ(ξt )S(t)I(t)(S(t) + I(t)) dt − σ(ξt )S(t)I(t)dB(t), dI(t) = β(ξ )S(t)I(t) − (µ(ξ ) + λ(ξ ))I(t) dt + σ(ξt )S(t)I(t)dB(t), t t t dR(t) = λ(ξt )I(t) − (µ(ξt ) + γ(ξt ))R(t) dt (3.19) 79 z In that paper, they showed that Theorem 3.3.1 We have the following assertions i) If β(j) > σ (j) and P πj C(j) < then the disease-free equilibrium is globally asymptotically stable in probability, where C(j) = β(j) − µ(j) − λ(j) − σ (j) P β (j) − 2µ(j)σ (j) ii) If πj < then the disease-free equilibrium is globally 2σ (j) asymptotically stable almost surely iii) If P πj C(j) > then the disease persists P On the other hand, for this model, our threshold λ is determined by λ = j∈S πj (β(j)− P πj C(j) an application of our results reads that if µ(j) − λ(j) − σ (j)) = P λ = πj C(j) < 0, the disease-free equilibrium is globally asymptotically stable P almost surely In case λ = πj C(j) > 0, the disease persists Thus, our findings provide sharper results for the extinction of the disease because we not need the additional condition that β(j) > σ (j) as in i) of Theorem 3.3.1 Moreover, since β(j) − σ (j) β (j) 2σ (j) (following Cauchy’s inequality β(j) β (j) 2σ (j) + σ (j) ) we have β (j) − 2µ(j)σ (j) β (j) − 2µ(j)σ (j) C(j) = β(j)−µ(j)−λ(j)− σ (j) −λ(j) < 2σ (j) 2σ (j) which shows that the condition in ii) of Theorem 3.3.1 is much more restrictive than ours In [90], the authors considered the model βS(t)I(t) dS(t) = Λ − µS(t) − + δR(t) dt − σS(t)I(t) 1+αI(t) 1+αI(t) dB(t), βS(t)I(t) dI(t) = − (µ + γ + ε)I(t) dt + σS(t)I(t) 1+αI(t) 1+αI(t) dB(t), dR(t) = γI(t) − (µ + δ)R(t)dt They proved that if ˜ := R βΛ σ Λ2 − > 1, µ(µ + γ + ε) µ (µ + γ + ε) then the system is persistent in time-average For (3.20), our threshold Λ σ Λ2 λ := β − (µ + γ + ε) − = (R0 − 1)(µ + γ + ε) µ µ 80 z (3.20) Thus, our theorem shows the persistence in probability which is stronger then persistence in time-average as λ > or equivalently R0 > Regarding to the extinction, we provide a more relaxing condition More specifically, our results read that if ˜ < 1) then the disease goes extinct with probability one, λ < (or equivalently R while the condition for extinction in [90] is either β2 ˜ < and σ βµ (a) σ > or (b) R 2(µ + γ + ε) Λ Clearly, under either (a) or (b), we have λ < (the inverse implication is not true), which implies that our condition for extinction λ < is more relaxing Furthermore, Lahrouz, Omari and Kiouach [46] proposed the model βS(t)I(t) S(t)I(t) dS(t) = b − µS(t) − + γR(t) dt − σ dB(t), + aI(t) + aI(t) βS(t)I(t) S(t)I(t) − (µ + c + α)I(t) + dI(t) = dt + σ dB(t) + aI(t) + aI(t) dR(t) = αI(t) − (µ + γ)R(t)dt They defined the reproduction number R0 = x2 + x3 < (3.21) βb , ∆ := x ∈ R3+ : x1 + µ(µ + c + α) b µ σ2 β2 b > , then the disease-free equilibrium state ( , 0, 0) is µ µ almost surely exponentially stable in ∆ Theorem 3.3.2 If Theorem 3.3.3 If R0 < 1, then I(t), R(t) converge almost surely exponentially to (0, 0) Theorem 3.3.4 If R0 < then S(t), I(t), R(t) converge almost surely exponentially b to ( , 0, 0) in ∆ µ βb − (µ + µ which obviously shows In constrast, our condition for exponentially fast extinction is simply λ = 2 2 b σ c + α) − b2µσ2 < which is equivalent to R0 < + 2µ2 (µ+c+α) that their conditions are more restrictive They have no conditions for persistence Our condition for persistence is simply λ > To illustrate our results, we consider the incidence rates are Beddington-DeAngelis functional responses; that is β1 (ξt ) F1 (S, I, ξt ) = , + a1 (ξt )S + b1 (ξt )I F2 (S, I, ξt ) = 81 z β2 (ξt ) , + a2 (ξt )S + b2 (ξt )I where , bi , βi , i = (1, 2) are positive constants The process {ξt , t > 0} is a right continuous Markov chain taking values in S = {1, 2} The transition rate from state to is q12 = 0.5 and the transition rate from state to is q21 = 0.8, then the , 13 ) stationary distribution π = (π1 , π2 ) = ( 13 Example 3.3.5 We assume that the capacity of the environment K = and the coefficients of the Equation (3.3) are given in Table 3.1 below Further, if ξt = 6, then we are in state and if ξt = then we are in state Coefficients a1 a2 b1 b2 β1 β2 γ1 γ2 µ ρ 1.5 0.1 1 0.5 2 0.3 1.2 2 0.2 1.5 0.2 States Table 3.1: Values of the coefficients in Example 3.3.5 By using formula (3.4), we have λ = −4.8819 < As a result of Theorem 3.2.2, I(t) tends to and S(t) tends to K as t → ∞ This claim is supported by Figure 3.1 That is, the population will eventually have no disease Sample path of I(t) I(t) xi(t) S(t) xi(t) 6 5 S(t) I(t) Sample path of S(t) 4 3 2 1 0 10 15 20 25 30 t 10 15 20 25 30 t Figure 3.1: Sample paths of I(t) (in blue on the left), S(t) (in blue on the right), and ξt (in red) in Ex 3.3.5 Example 3.3.6 We assume that the capacity of the environment is K = 30 In case ξt = 1, then we are in state and if ξt = then we are in state The Table of parameter values is given below 82 z Coefficients a1 a2 b1 b2 β1 β2 γ1 γ2 µ ρ 1.7 0.5 1.2 0.7 0.5 0.7 0.6 1.5 1.5 0.5 0.5 States Table 3.2: Values of the coefficients in Example 3.3.6 Detailed computations give us that λ = 2.3441 > Thus, by Theorem 3.2.3, I(t) is strongly stochastically persistent That result can be observed in Figure 3.2 The densities of the empirical measure, which approximate the invariant densities, are shown in Figure 3.3, Figure 3.4 in 2D and 3D settings respectively Sample path of I(t) 15 Sample path of S(t) 30 I(t) xi(t) S(t) xi(t) 25 20 I(t) S(t) 10 15 10 0 10 15 20 25 30 35 40 45 50 10 15 20 25 t 30 35 40 45 50 t Figure 3.2: Sample paths of I(t) (in blue on the left) and S(t) (in blue on the right) and ξt (in red) in Ex 3.3.6 #10 graph of density p(i;s;1) in 2D #10 graph of density p(i;s;2) in 2D 14 14 4.5 2.5 12 12 3.5 10 10 8 2.5 s s 1.5 6 1.5 4 0.5 2 0.5 0 0 10 15 20 25 0 i 10 15 20 25 i Figure 3.3: Empirical density of (S(t), I(t)) with respect to the first 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 3.3.6 in 2D 83 z graph of density p(i;s;1) in 3D graph of density p(i;s;2) in 3D #10 #10 p(i,s) 4 p(i,s,1) 30 15 20 15 10 10 10 s i s 0 10 15 20 25 30 i Figure 3.4: Empirical density of (S(t), I(t)) with respect to the first 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 3.3.6 in 3D Conclusion of Chapter This chapter sets out a stochastic SIRS model with regime switching In this model, we have determined a threshold value whose sign specifies whether or not the disease goes to extinct or survive permanently Working with a general incidence rate and a taking into account both white noise and color noise, the model includes almost all SIRS models appeared in the literature (e.g [8, 29, 46, 77, 90]) In this chapter, a nearly full classification for the asymptotic behaviors of the model has been given Only the critial case when λ = is not studied yet We also provide the exact exponential convergence rate when λ < which is not obtained using existing methods In constrast, in most existing results, besides a threshold, some additional conditions are needed in order to obtain the extinction and/or the permanence of the disease As a result, our findings can be seen as significant extensions of results in the aforementioned papers 84 z Conclusion In this thesis, we have studied effect of the random perturbations on three biological systems: the predator-prey systems in Chapter 2, and the SIRS epidemic evolution in Chapter Our main results of this thesis can be summarized as follows • We established the conditions for the existence and uniqueness of global solutions (Theorems 2.1.1, 2.2.1 and Theorems 3.2.1) • We provided the threshold values to classify the systems (Formulations (2.15), (2.55) and formulation (3.4)) This allows us to derive not only sufficient conditions but also almost necessary conditions for the permanence (Theorems 2.1.2, 2.2.7, 3.2.2 and Theorem 3.2.3) • We estimated the convergence of x(t) to their solutions on the boundary equations (Theorem 2.1.2 and Theorem 2.2.7) As a result, our finding in [Pub 1] can be seen as a generalization and an improvement of Dang, Du and Ton in [15] • We obtained the conditions for the existence of the stationary distributions and for the validity of the strong law of larger numbers (Theorems 2.1.2, 2.2.7, 2.1.4 and Theorem 2.2.8) Future research In the future, we would like to consider the critical case λ = in some particular cases Moreover, we will focus on hybrid switching diffusion systems with Lévy noise Also, we plan on considering the class of regime switching diffusion systems with switching states x-dependent generator One of my goals is to establish a classification for extinction and persistence in these models For these problems, we expect to encounter more challenging aspects 85 z The author’s publications related to the thesis [P ub 1] N H Du, N T Dieu, T D Tuong (2017), “Dynamics behavior of a stochastic predator-prey system under regime switching”, Discrete Contin Dyn Syst Ser B 22 (9), pp 3483-3498 (SCI, for Chapter of this thesis) [P ub 2] T D Tuong, N H Dang, N T Dieu, T Q Ky (2019), “Extinction and permanence in a stochastic SIRS model in regime switching with general incidence rate”, Nonlinear Anal Hybrid Syst 34, pp 121-130 (SCIE, for Chapter of this thesis) [P ub 3] T D Tuong, N T Dieu, N H Du, “On the asymptotic behavior of a stochastic model with Ivlev’s functional response and jumps”, Stochastic Models, revised (for Chapter of this thesis) 86 z 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 dxi (t) = xi (t) ri (ξt ) + n X ! aij (ξt )xj (t) dt + αi (ξt )xi (t)dB1 (t) j=1 + n X σij (ξt )xi (t)xj (t)dB2 (t), i = 1, , n (1) j=1 For any constant sequence {cij (k)}, (1 i, j n, k ∈ S), define cu = max 16i,j6n,k∈S cij (k), cl = 16i,j6n,k∈S cij (k) cu (k) = max cij (k), cl = cij (k) 16i,j6n 16i,j6n Theorem A.1 ([53, Theorem 7]) If for each u ∈ S, σii (u) > for i n whilst σij (u) > for i 6= j, the solution x(t) of (1) obeys ln |x(t)| 1, a.s t→+∞ ln t Pn Proof Define W (x) = ln U (x) = ln i=1 xi By using the generalized Itô’s formula, lim sup 87 z we have ! n n X X xi ri (ξ) + aij (ξ)xj dt dW (x) = U (x) i=1 j=1 " #2 #2 " n n n XX X − σij (ξ)xi xj dt αi (ξ)xi + 2U (x) i=1 j=1 i=1 n n n X XX + αi (ξ)xi dB1 (t) + σij (ξ)xi xj dB2 (t) U (x) i=1 U (x) i=1 j=1 Applying the generalized Itô’s formula again, we obtain " n !# n n X X X d(exp(t)W (x)) = exp(t) ln xi + xi ri (ξ) + aij (ξ)xj dt U (x) i=1 i=1 j=1 " #2 # " n X n n X X exp(t) σ (ξ)x x dt α (ξ)x + − ij i j i i 2U (x) i=1 j=1 i=1 n n n exp(t) X exp(t) X X + αi (ξ)xi dB1 (t) + σij (ξ)xi xj dB2 (t) U (x) i=1 U (x) i=1 j=1 Thus, we have already shown that exp(t) ln n X xi (t) − ln i=1 t Z = " exp(s) ln + U (x(s)) n X xi (0) i=1 n X xi (s) i=1 n X n X i=1 j=1 xi (s) ri (ξs ) + !# aij (ξs )xj (s) " #2 n X exp(s) αi (ξs )xi (s) − 2U (x(s)) i=1 " n n #2 XX + σij (ξs )xi (s)xj (s) ds + N (t), ds t Z i=1 j=1 where Z t n exp(t) X αi (ξs )xi (s)dB1 (s) N (t) = U (x(s)) i=1 Z t n n exp(s) X X + σij (ξs )xi (s)xj (s)dB2 (s) U (x(s)) i=1 j=1 88 z (2) The quadratic form of N (t) is " #2 n exp(2s) X αi (ξs )xi (s) hN (t), N (t)i = U (x(s)) i=1 #2 " n n XX ds σij (ξs )xi (s)xj (s) + Z t i=1 j=1 By the exponential martingale inequality, we obtain that P sup [N (t) − 0.5ε exp(−τ k)hN (t), N (t)i] > ε−1 ρ exp(τ k) ln k k −ρ , 06t6τ k 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), N (t) 0.5ε exp(−τ k)hN (t), N (t)i + ε−1 ρ exp(τ k) ln k, t τ k In other words, " n #2 exp(2s) X N (t) 60.5ε exp(−τ k) αi (ξs )xi (s) ds U (x(s)) i=1 " n n #2 Z t exp(2s) X X σij (ξs )xi (s)xj (s) ds + 0.5ε exp(−τ k) U (x(s)) i=1 j=1 Z t + ε−1 ρ exp(τ k) ln k for t τ k Substituting this inequality into Equation (2) gives exp(t) ln n X i=1 t Z = xi (t) − ln n X xi (0) i=1 " exp(s) ln + U (x(s)) n X xi (s) i=1 n X n X i=1 j=1 xi (s) ri (ξs ) + !# aij (ξs )xj (s) ds #2 n X exp(s) αi (ξs )xi (s) [1 − ε exp(s − τ k)]ds − 2U (x(s)) i=1 " n n #2 Z t exp(2s) X X − σij (ξs )xi (s)xj (s) [1 − ε exp(s − τ k)]ds U (x(s)) i=1 j=1 Z t " + ε−1 ρ exp(ξk) ln k (3) 89 z Since s τ k, then #2 " n Z t X exp(s) − αi (ξs )xi (s) [1 − ε exp(s − τ k)]ds < 2U (x(s)) i=1 At the same time U (x) n X xi ri (ξ) + n X ! u |r| + |a| aij (ξ)xj j=1 i=1 u n X xi i=1 Moreover, there exists a positive constant ν such that #2 " n n Z t X X exp(2s) σij (ξs )xi (s)xj (s) [1 − ε exp(s − τ k)]ds U (x(s)) i=1 j=1 #2 " n Z t X exp(s) xi (s) [1 − ε exp(s − τ k)]ds >ν i=1 Then it follows from (3) that n n X X xi (0) xi (t) − ln exp(t) ln i=1 Z t i=1 n X " exp(s) ln − ν " u xi (s) + |r| + |a| u i=1 n X xi (s) i=1 #2 xi (s) n X [1 − ε exp(s − τ k)] ds 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) t τ k and k > k0 (ω), we have ln n X xi (t)/ ln t exp(−t) ln i=1 n X xi (0)/ ln t 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 Pn 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 i n; σij > 0, if i 6= j Then 90 z (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 "Z n # tX lim sup E x2+θ (s)ds Kθ a.s i t t→∞ i=1 n,o Proof Define a C -function V : Rn,o + → R+ by V (x) = Pn θ i=1 xi According to the Itô’s formula, ! n n n X X 1X θxi bi + aij xj + dV (x(t)) = θ(θ − 1)xθi i=1 j=1 i=1 + n X θxθi n X i=1 n X !2 xj σij dt j=1 σij xj dB(t) j=1 Moreover, it is easy to show that n X θxi bi + n X ! aij xj n X and n X n X θ)xθi θ(1 − i=1 !2 σij > j=1 As a result, we obtain " dV (x(t)) θ n X |bi |xi + i=1 n X +θ xθi i=1 n X θ(1 − θ)x2+θ σii2 i i=1 # n θ(1 − θ) X 2+θ |aij |xi xj − σii xi dt j=1 i=1 n X n X i=1 n X |aij |xi xj i=1 j=1 i=1 j=1 i=1 θxi |bi | + n X n X σij xj dB(t) (5) j=1 Furthermore, by taking into consideration the fact that the polynomial θ n X |bi |xi + i=1 n X n X i=1 n θ(1 − θ) X 2+θ |aij |xi xj − σii xi j=1 i=1 has an upper positive bound, say Kθ , inequality (5) yields Z n Z t θ(1 − θ) t X 2+θ V (x(t)) + σii xi V (x(0)) + Kθ ds + M (t), i=1 where M (t) = θ R t Pn θ i=1 xi Pn j=1 σij xj dB(s) (6) is a real-valued continuous local mar- tingale vanishing at t = Taking expectations on both sides of (6), this completes the proof 91 z 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) = EY, d2 u dθ2 (θ) K2 , θ ∈ [0, 0.5] for some K2 > depending only on K1 Thus, it follows from Taylor’s expansion that u(θ) EY θ + K2 θ2 , θ ∈ [0, 0.5] 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 ... 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 MÔI TRƯỜNG NGẪU NHIÊN Chun ngành: Phương trình Vi phân Tích phân Mã số: ... tắt Tóm tắt Trong luận án nghiên cứu dáng điệu tiệm cận lớp mơ hình sinh thái, mơ tả phương trình vi phân ngẫu nhiên, để mơ tả tương tác thú mồi nghiên cứu lan truyền dịch bệnh Đối với hệ thú-mồi,... 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,
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