Dáng điệu tiệm cận của một số mô hình quần thể trong hệ sinh thái với môi trường ngẫu nhiên

VIETNAM NATIONAL UNIVERSITY, HANOIVNU UNIVERSITY OF SCIENCE Tran Dinh Tuong ASYMPTOTIC BEHAVIOR OF POPULATION MODELS IN ECOSYSTEM WITH RANDOM ENVIRONMENT Speciality: Differential and Int

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 OFDOCTOR 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

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 TS NGUYỄN HỮU DƯ PGS TS NGUYỄN THANH DIỆU

HÀ NỘI – 2020

Page

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 2 Long-term behavior of stochastic predator-prey systems 34 2.1 Dynamic behavior of a stochastic predator-prey system under regime switching 35

2.1.1 Introduction 352.1.2 Sucient and almost necessary condition for permanence 362.1.3 Discussion and numerical solutions 512.2 On the asymptotic behavior of a stochastic predator-prey model withIvlev's functional response and jumps 552.2.1 Introduction 552.2.2 Introductory results 572.2.3 Almost necessary and sucient condition for extinction and

permanence 612.2.4 Discussion and numerical example 67Chapter 3 Extinction and permanence in a stochastic SIRS model

in regime switching with general incidence rate 713.1 Introduction 713.2 Sucient and almost necessary conditions for permanence 733.3 Discussion and numerical experiments 79

The author's publications related to the thesis 85

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 myprimary supervisor, Professor Nguyen Huu Du for his tireless support and endlessguidance, and infusing spirit into my research

I also wish to express the deepest thanks to the thesis co-supervisor, Associate fessor Nguyen Thanh Dieu for his kind comments, valued suggestions, and sharinggreat ideas during this course

Pro-I am particularly grateful to Dr Nguyen Hai Dang for many valuable discussionsand his great support during my work I own my thanks to Dr Tran Quan Ky forhis contribution to our joint work [Pub 2]

I would like to thank all the teachers, staff members and the management of theFaculty of Mathematics Mechanics and Information Technology at VNU University

of Science, Vietnam National University, Hanoi as well as VIASM for their greatsupport, lessons as well as opportunities for completion of the research

I wish to thank my friends, who always trust, encourage and support me over theyears 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 candidateTran Dinh Tuong

Abstract In this thesis, we consider long-term behavior of a class of formulatedpopulation models using stochastic differential equations to describe predator-preyrelationships 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’sfunctional response perturbed simultaneously by the white noise and Lévy noise Forthe 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 Wepropose new approaches to provide thresholds which indicate whether the systemsare eventually extinct or permanent This allows us to derive not only sufficientconditions but also almost necessary conditions for permanence (as well as ergodic-ity) based on sign of such thresholds Furthermore, conditions for the existence ofstationary distributions and for the validity of the strong law of large numbers areestablished 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

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 Đốivới các hệ thú-mồi, chúng tôi nghiên cứu hai mô hình: mô hình thứ nhất với nhiễutrắng và nhiễu màu, mô hình thứ hai có đáp ứng chức năng dạng Ivlev bị chịu đồngthời 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 bệnhtổng quát Chúng tôi đề xuất các phương pháp tiếp cận mới để xây dựng các giá trịngưỡng 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 ergodiccủa hệ Hơn nữa, các điều kiện cho sự tồn tại của các phân phối dừng cũng như cácđiều kiện cho luật số lớn có hiệu lực được thiết lập trong các trường hợp cụ thể

Từ khóa 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 chứcnăng dạng Ivlev, sự tuyệt chủng, sự tồn tại bền vững, quá trình khuếch tán có bướcnhảy, mô hình SIRS, các mô hình dịch bệnh

This work has been completed at VNU University of Science, Vietnam NationalUniversity, 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 areallowed by my coauthors to be presented in this thesis, are new and they have neverbeen used in any other theses

Author

Tran Dinh Tuong

List of Figures

2.1 Trajectories of y(t) in the state 1 (blue line) and in the state 2 (redline) in Ex 2.1.6 532.2 A switching trajectory y(t) in Ex 2.1.6 on the left and A switchingtrajectory y(t) in Ex 2.1.7 on the right 532.3 Trajectories of y(t) in the rst state (blue line) and the second state(red line) respectively in Ex 2.1.7 542.4 Phase picture and empirical density of x(t), y(t)
in Ex 2.1.7 in 2Dand 3D settings respectively 552.5 Phase picture of (x(t), y(t)) and empirical density of invariant mea-sure settings respectively (with λ1 = 1) Dierent colors representdierent sizes of the density 682.6 Trajectories of x(t) on the left and of y(t) on the right with intensity

λ1 = 6.667 692.7 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 3.3.5 823.2 Sample paths of I(t) (in blue on the left) and S(t) (in blue on theright) and ξt (in red) in Ex 3.3.6 83

3.3 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 aninvariant probability measure of Ex 3.3.6 in 2D 833.4 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 aninvariant probability measure of Ex 3.3.6 in 3D 84

List of Tables

2.1 Values of the coecients in Example 2.1.6 52

2.2 Values of the coecients in Example 2.1.7 54

3.1 Values of the coecients in Example 3.3.5 82

3.2 Values of the coecients in Example 3.3.6 83

List of Notations

i.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

Rn,o+ {(x1, , xn) ∈ Rn : x1 > 0, , xn > 0}

Rn+ {(x1, , xn) ∈ Rn : x1 > 0, , xn > 0}

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

S {1, 2, , 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,

i.e trace A =P

16i6nAii

A> the transpose of a vector or matrix A

|x| the Euclidean norm of a vector x

Lp

([a, b]; Rn) the family of Rn-valued Ft-adapted processes {f (t)}a6t6b

such that Rab|f (t)|pdt < ∞ a.s

of specialized research: computer models and automata theory, mathematical physics, molecular set theory, etc and it tends to be helpful in both theoretical andpractical research.

bio-This thesis belongs to the field of mathematical biology Although there are manyways 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) Bymodeling method, ODEs allow us to test and generate hypotheses, and help usdescribe the evolution in times of population That is the simplest way to study thedevelopment of a population viewed as a dynamical system

The investigation of population change by ODEs has a long term history In hisbook [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 inturn restored the original per capita production level, and he proposed the simplestpopulation of single species Observing that there must be adjustment to such ex-ponential growth, in the long run of course, Verhulst [81] introduced the logisticmodel In order to study of the interrelationship between species and their envi-

ronment, in such areas as predator-prey and competition interactions, in [57] Lotkastudied autocatalytic and the theory of competing species which was later followed

by Volterra blue by proposing the simplest Lotka-Volterra model also known asthe 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 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

Kol-to show the likely outcome of an epidemic and help inform public health tions The research results are helpful to predict the developing tendency of theinfectious disease, to decide the key elements of the spread of infectious disease and

interven-to seek the optimum strategies of anticipating and controlling the spread of tious diseases By computer simulations with dynamic methods, this theory couldmake modeling and the original analysis more realistic and more reliable, make thecomprehension for spread rule of infectious diseases more thorough The earliestaccount of mathematical modeling of spread of disease was carried out in 1766 byBernoulli [30] who was one of the first mathematicians to attempt to model theaffect of the disease in a population, on inoculation against smallpox Another valu-able contribution to the understanding of infectious diseases is a result of Simon in

infec-a outbreinfec-ak of cholerinfec-a thinfec-at occurred in 1854, Englinfec-and infec-and occurred during the

1846-1860 cholera pandemic happening worldwide An early reference of epidemic models

is made in a paper by Hamer [7] In his work, the probability of an infection in thenext period of time was proportional to the number of infectious individuals multi-plied by the number of susceptible individuals This principle was called the massaction and has been used in many areas of science, in particular to determine therate of chemical reactions Now, the popular epidemic dynamic models are still socalled compartmental models which were constructed by Kermack and Mckendrick[39] and have been developed by many scholars Their model, nowadays best known

as the SIR model, the population is divided into three compartments: susceptiblecompartment (S), in which all individuals are susceptible to the disease; infected

compartment (I), in which all individuals are infected by the disease and have fectivity; removed compartment (R), where all the individuals recovered from theclass (I) and have permanent immunity However, some removed individuals maylose immunity and return to the susceptible compartment This situation can bemodeled by an SIRS model (see [32, 67, 68, 69, ]).

in-In order to study these models, some classical methods come from the theory of namical systems by determining system’s equilibrium points, and using linearizationtechniques to describe the behavior of the system near the equilibria The main tools

dy-in studydy-ing the stability are Lyapunov functions, Lyapunov exponents, etc over, numerical experiments of specific models will be conducted to illustrate thetheoretical findings ([64, 67, 68, ]) It has been well recognized that the traditionalmodels are often not adequate to describe the reality due to random environment andother random factors For example, in an ecology system the growth rates and thecarrying capacities are usually changed due to environmental noise These changesusually cannot be described by the traditional deterministic population models Forinstance, the growth rates of some species in the rainy season will be much differentfrom those in the dry season Note that the carrying capacities often vary accord-ing to the changes in nutrition and/or food resources Likewise, the interspecific orintraspecific interactions differ in different environments Similarly, many randomfactors effect epidemic Cold and flu are influenced by humidity and cold temper-atures Viruses are more likely to survive in cold and dry conditions Hard winds,rain, cold as well as large variations of temperatures are factors that weaken theimmune system Lack of sunlight also provokes a decrease of the level of D vitamin.Clearly, the environment and weather changes cannot be modeled as solutions ofdifferential equations in the traditional setup For this reason, it is more realistic totake into account the impacts of random noises This demand has been stimulatingthe developments of stochastic models in biology for the past decades Recently,resurgent attention has been drawn to treat stochastic systems in which the ran-dom noise is formulated in terms of some random noises In this case, stochasticdifferential equations (SDEs) become a useful tool to study this phenomenon withmany books or papers dealing with this field (see [2, 14, 23, 31, 61, 62, 71, ] andreferences therein)

More-To make the models closer to reality, we need to consider the random perturbations

of the environment on the population As an illustration, the distinctive seasonalchange such as dry and rainy seasons can be affected to the population These ef-fects of environment regimes in memoryless conditions to population are called colornoise and can be illustrated as a Markovian switching (also known as regime switch-ing) between two or more regimes of environment This situation can be explained

by a hybrid switching diffusion process in which its environment is perturbed byboth Brownian motion and Markovian switching Actually, the term “hybrid” sig-nifies the coexistence of continuous dynamics and discrete events in the system.The hybrid switching diffusions have a wide range application: wireless communica-tions, signal processing, queueing networks, production planning, biological systems,ecosystems, financial engineering, and modeling, analysis, and control and optimiza-tion of large scale systems Therefore, the systems have become a hot topic in recentyears [13, 14, 17, 29, 47, ] For instance, Han and Zhao [29] studied the stability

of a SIRS model under regime switching In 2006, Mao and Yuan [62] presentedcomprehensive treatments of switching diffusion processes in this textbook Thetextbook covered various types of stochastic differential equations with Markovianswitching and emphasized the analysis of stability of the systems In [86], Yin andZhu studied the stability of diffusion processes with state-dependent switching Thistextbook also constructed algorithms to approximate solutions of certain systems,and provided sufficient conditions for convergence for numerical approximations tothe invariant measures The results presented in the books [62] and [86] are useful toresearchers working in stochastic modeling, systems theory, and applications wherecontinuous dynamics and discrete events are intertwined

Another phenomenon is that, the biological systems may suffer sudden mental shocks: earthquakes, tidal waves, tropical storms, surges, etc When therandom perturbations occur, the population sizes may change unexpectedly As aconsequence, the frameworks turn out to be very perplexing, the sample paths arediscontinuous and the above models cannot clarify such phenomena To explainthese phenomena, a mixture model mixing a jumps process and a diffusion processwhich is perturbed by white noise, are considered This model can be considered

environ-as involvement of the deterministic part and the random part including jumps and

has gained much attention [2, 5, 50, 56, 91, ] We refer to the readers to [71]for providing of preliminaries of stochastic calculus with Lévy processes: Itô-Lévydecomposition, Itô formula, etc Rong [74] introduced the concept of solutions anddiscussed their existence and uniquess and the related important theory: matin-gale representation theorem, finding the solutions of backward SDEs and filteringproblem Siakalli [78] studied the stabilizing effects of the Lévy noise in the system,proved the existence of sample Lyapunov exponents of the trivial solution of thestochastically perturbed system, and provided sufficient criteria under which thesystem was almost surely exponentially stable This model has important applica-tions in magnetic reconnection, coronal mass ejections, condensed matter physics,

in pattern theory and computational vision, particularly in economics

Some natural questions arise when we study the behavior of stochastic biologicalsystems For example, whenever the species in population will survive forever, howabout long-term asymptotic behaviors of the population and what are conditionsfor the extinction and/or persistence? A lot of works deal with these problems.For instance, Dang, Du and Ton in [15] developed result of Rudnicki [75] to astochastic predator-prey system where Brownian motion acts on the coefficients ofenvironment They also showed that the density functions of the solutions exist andthen, studied the asymptotic behavior of these densities In [63], Mao, Sabanis andRenshaw studied a stochastic Lotka-Volterra model by means of Lyapunov-typefunctions and martingale inequalities These useful tools have been also employedextensively for estimating the growth rates, and the average in time of species invarious stochastic systems (see [24, 75, ]) Moreover, several attempts have beenmade to provide conditions for extinction, permanence or stability of the systems([37, 54, ]) However, it is known that using Lyapunov function method, one canonly deduce the sufficient conditions Therefore, the conditions dealt with in theabove mentioned works are rather restrictive and not close to a necessary condition

As a consequence, they are unable to classify the evolution of all stochastic systems.Thus, although interesting, their works left a sizable gap

To fill this gap, Du, Dang and Yin [23] considered a stochastic predator-prey modelwith Beddington-DeAngelis functional response by using a quite new technique

To provide a sufficient and almost necessary condition for permanence (as well asergodicity), they constructed a threshold parameter Based on value of the threshold,

it is known that whenever the system is permanent or some species in the populationare extinct

In continuing to work in this field, the main first part of this thesis (Section 2.1 inChapter 2) consider long-term behavior of predator-prey models in the case that theenvironment is perturbed by random noises like white noise or Poisson type noise orboth Using technique in [23], we are able to derive sufficient and almost necessaryconditions for permanence (as well as ergodicity) and extinction of these models viathreshold parameters Furthermore, convergences in total variation norm of transi-tion probabilities to invariant measures are expected The first part is showed in thefirst section of this introduction as follows

1 Dynamics behavior of stochastic predator-prey systems

There are two stochastic predator-prey models that are studied in the first part

of the thesis The first system deals with a regime switching predator-prey modelperturbed by white noise In 2011, Dang, Du and Ton [15] studied a stochasticpredator-prey model







dx(t) = x(t) a1− b1y(t) − c1x(t)
dt + σx2(t)dB1(t),dy(t) = y(t) − a2+ b2x(t) − c2y(t)
dt + ρy2(t)dB2(t)

(0.1)

where x(t), y(t) are respectively the density of prey and predator populations, rameters: a1, b1, c1, σ, a2, b2, c2, ρ are positive numbers, B1(.), B2(.) are standard Brow-nian motion In that paper, the authors studied further the asymptotic behavior ofsystem (0.1) by considering the convergence of the density of the solution The prob-lem is much more complicated than those in Arnold, Horsthemke and Stucki [3].Because Khasminskii function method is not suitable to this model, the author ofpaper [15] studied further the asymptotic behavior the system by using the method

pa-of analyzing the boundary distributions More precisely, they concluded that

• In case without the preys, the predators die out with probability one

• In the absence of the predators, the quantity of the preys oscillates between 0and ∞ and the boundary equation has a unique stationary distribution with

the density f∗(x) Moreover, ln x(t) converges in distribution to f∗ as t → ∞.

The second evolution in the main first part of the thesis (Section 2.2 in Chapter 2)comes from a deterministic predator-prey which was introduced by Kooij [44]







dx(t) = x(t)[a1− b1x(t)] − c1y(t)[1 − e−γx(t)]
dtdy(t) = y(t) − a2+ c2[1 − e−γx(t)]
dt,

where x(t), y(t) denote the prey and the predator densities at time t respectively.Parameters a1, b1, a2, c1, c2 are positive real numbers, 1 − e−γx(t) is Ivlev’s functionresponse In that paper, the uniqueness of limit cycles for a predator-prey systemwith a functional response of Ivlev type is proved and more properties of the limitcycles are considered Following the method in [23], we will establish a sufficient andalmost necessary condition for permanence of a predator-prey model perturbed si-

multaneously by the white noise and Lévy noise with the Ivlev’s functional response

By concentrating on systems (0.2) and (0.3), our tasks are as follows:

• Obtaining the conditions for the existence and uniqueness of global solutions

of the equations

• Providing the threshold values to classify the asymptotic behavior these tems In the sequel, we aim to provide not only sufficient conditions but alsoalmost necessary conditions for permanence

sys-• Improving the estimation of convergence of x(t) to its solution on the boundaryequation (In [15], the convergence is convergence in distribution)

• Establishing the conditions for the existence of the stationary distributions andfor the validity of the strong law of larger numbers for the models

Besides, in the mathematical modeling of disease transmission, SIRS epidemic els can be seen as generalisations and to be more suitable than other simple Kermack-Mckendrick types Therefore an SIRS model will be proposed in the second part ofthe thesis Because the disease transmission process is unknown in detail, severalauthors proposed different forms of incidences rate in order to model the diseasetransmission process under taking into account the presence of white noise, colornoise and both of them For example, authors in [45] studied a deterministic SIRSmodel which has been extended to stochastic SIRS models (see [10, 29, 47, 77]),with the standard bilinear incidence rate N’zi and Tano [70] investigated stability

mod-of an SIRS epidemic model with a saturated incidence rate Yousef and Salman [87]studied a fractional-order SIRS epidemic model with a nonlinear incidence rate, etc

Motivated by the works, Chapter 3 of the thesis will consider asymptotic behavior of

a stochastic SIRS model in regime switching with a general incidence rate Results

of this chapter will be presented in the second section of this introduction

2 Extinction and permanence in a stochastic SIRS model in regime switching withgeneral incidence rate

One of our objectives in this chapter is to show that the stochastic models for tious diseases can be treated by a different approach that can work for more generalmodels rather than treating concrete models one by one with a common approachthat does not often work well Working on each model, the common approach is

infec-to have global estimates using limit theorems of martingales and some algebraicinequalities to find conditions for extinction Besides, in order to find conditions forpersistence, the technique to introduce a Lyapunov function V and impose someconditions such that the Lyapunov function satisfies some desired properties such

as LV < 0 outside a compact set However, this method does not provide goodinsight into the dynamics of our model because the Lyapunov function is chosensubjectively rather than based on the nature of the dynamic system Choosing dif-ferent functions gives different sets of conditions We will treat this problem by theapproach that was introduced by Du, Dang and Yin in [23] The main idea of thisapproach is that the persistence and extinction of a system in the interior dependstrongly on the dynamics near the boundary Thus, we analyze the dynamics onthe boundary, namely, stationary distributions on the boundary and check if theyare “repellers” or “attractors” Intuitively, if all the stationary distributions on theboundary are “repellers”, then the system is persistent Otherwise, extinction wouldhappen The results obtained using the approach are therefore very sharp and onlyleave “critical cases” unsolved As a main goal of this part, we drive a threshold valuethat which determines whether the disease is extinct or permanent Our model isthe following stochastic differential equation

+S(t)I(t)F2(S(t), I(t), ξt)dB(t)dR(t) = (γ2(ξt)I(t) − (µ(ξt) + γ1(ξt))R(t))dt,

(0.4)where S(t), I(t), R(t) are susceptible, infected, removed classes respectively, ξt, t > 0

is a right continuous Markov chain taking values in S = {1, 2, , N }, F1(·), F2(·)are positive and locally Lipschitz functions on [0, ∞)2× S, B(t) is a one dimensionalBrownian motion, all parameters K, µ(i), ρ(i), γ1(i), γ2(i) are assumed to be positivefor all i ∈ S By analyzing the dynamics on the boundary of Equation 0.4, ourcontributions in Chapter 3 are as follows:

• Establishing the conditions for the existence and uniqueness of global solutions

of Equation 0.4

• Providing a threshold parameter to classify the systems Based on sign of thethreshold, we obtain not only sufficient condition but also almost necessarycondition for permanence of the disease

The thesis is organized as follows Chapter 1 presents some basic knowledge aboutstochastic processes as martingale process, Markov process, Lévy process and thetheory of SDEs which includes SDEs with Markovian switching and SDEs withjumps After that some notations and definitions for ecosystem and relative fieldsare given Chapter 2 and 3 are the core of the thesis Chapter 2 deals with twostochastic predator-prey models In the first model, we present a regime switchingpredator-prey model perturbed by white noise We give a threshold by which weknow whenever a switching predator-prey system is eventually extinct or permanent

We also give some numerical solutions to illustrate that under the regime switching,the permanence or extinction of the switching system may be very different from thedynamics in each fixed state In the second system of Chapter 2, we study long-timebehavior of a stochastic predator-prey model with Ivlev functional response effected

by both white noise and Lévy jumps We focus on conditions for the existence ofthe ergodic stationary distribution to the logistic equation Next, by introducing thethreshold λ, sufficient and almost necessary conditions for the permanence as well

as ergodicity and extinction are investigated: when λ < 0, the species converge tothe population on the boundary in exponential rate If λ > 0, there exists uniqueinvariant probability measure concentrated on an appropriate space

Chapter 3 devotes to study a SIRS epidemic model with general incidence rate Weconsider a stochastic SIRS model with general incidence rate and perturbed by boththe white noise and color noise As in Chapter 2, we also determine the threshold λthat is used to classify the extinction and permanence of the disease In particular,

λ < 0 implies that the disease-free (K, 0, 0) is globally asymptotically stable, i.e.,the disease will eventually disappear If λ > 0 the epidemic is strongly stochasticallypermanent Comparison to the literature and numerical simulations to demonstrateour results are presented to end this chapter Finally, some supplements provideadditional information in Appendices section and Conclusions are given to end thethesis

Results of this thesis have been presented in seminars, workshops, and several ferences:

con-• Vietnam-Korea Workshop on Dynamical System and Applications, Hanoi 2016

• 15th Workshop on Optimization and Scientific Computing, Ba Vi 2017

• Seminar at Faculty of Basic Sciences, Ho Chi Minh University of Transport, HoChi Minh 2018

• 9th Vietnam Mathematical Congress, Nha Trang 2018

• International Workshop on Probability Theory and Related Fields, Hanoi 2019

• Summer School on Data and Models in Ecology and Evolution, Paris 2019

Chapter 1

Preliminaries

This introductory chapter begins by providing necessary background information

to contextualise the thesis This can be seen as a prerequisite This chapter mainlysurveys on the stochastic processes, the theory of stochastic differential equations(SDEs) Firstly, some definitions and notions in stochastic calculus such as mar-tingale processes, Markov processes, Lévy processes and related results are given.Next, the chapter also presents SDEs with Markovian switching and one with jumps,which are useful tools to study all models in this thesis Finally, some preliminariesfor stochastic mathematical models in ecosystem are given The basic theorems inthis chapter are stated without proofs We can find proofs in the references

1.1 Stochastic processes

Denoting Rn,◦+ := (0, ∞)n, Rn+ := [0, ∞)n, in case n = 1 we shall omit parameter

n in the sets Let (Ω, F , P) be a probability space A filtration is a family {Ft}t>0

of increasing sub-σ-algebras of F The filtration is called to be right continuous if

Ft = ∩s>tFs for all t > 0 When the probability space is complete, the filtration

is called to satisfy the usual hypotheses if it is right continuous and F0 containsall the P-null sets From now on, unless otherwise specified, we shall always work

on a given complete probability space (Ω, F , P) with a filtration {Ft}t>0 satisfyingthe usual conditions Let I be a subset of R, and Ξ be a metric space with Borelσ-algebra B(Ξ) A family random variables {Xt}t∈I takes value in Ξ is said to be a

stochastic process with parameter set I and state space Ξ For each fixed t ∈ I, wehave a random variable Xt(ω) On the other hand, for each fixed ω ∈ Ω, we have afunction t 7→ Xt(ω) ∈ Ξ that is called a sample path (or trajectory) of the process.Throughout of this thesis, we only deal with the parameter set I = [0, ∞).

Let {Xt}t∈R+ be a stochastic process taking values in Ξ The process is said to

be càdlàg (right continuous and left limit) if it is right continuous and for all mostall ω ∈ Ω the left limit lims↑tXs(ω) exists and is finite for all t > 0 It is said to

be continuous (right continuous, left continuous) if for almost all ω ∈ Ω, function

Xt(ω) is continuous (right continuous, left continuous) on R+ respectively It is said

to be {Ft}-adapted if for every t, Xt is Ft-measurable It is called to be measurable

if the stochastic process regarded as a function of two variables (t, ω) from R+× Ω

to Ξ is B(R+) × F -measurable LetP denote the smallest σ-algebra on R+× Ω withrespect to which every left continuous process is a measurable function of (t, ω) Astochastic process is said to be predictable if the process regarded as a function of(t, ω) is P-measurable Let {Xt}t∈R+ be a stochastic process, any stochastic process{Yt}t∈R+ is said a modification of {Xt} if for all t > 0, Xt = Yt a.s Here is a class ofuseful stochastic processes for developing some results in this thesis

1.1.1 Martingale process

A random variable τ : Ω → [0; ∞] is said an {Ft}-stopping time (stopping time, forthe sake of simplicity) if {ω : τ (ω) 6 t} ∈ Ft for any t ∈ R+ If τ is a stoppingtime, let Fτ = {A ∈ F : A ∩ {ω : τ (ω) 6 t} ∈ Ft, ∀t > 0} that is a sub-σ-algebra of F A real-valued {Ft}-adapted integrable process {Mt}t>0 is called amartingale with respect to {Ft} (or short, martingale) if E|Mt| < ∞ and E(Mt|Fs) =

Ms a.s for all 0 6 s < t < ∞ We can always assume that any martingale is càdlàg

A real-valued {Ft}-adapted integrable process {Mt}t>0 is said submartingale withrespect to {Ft} (shortly submartingale) if E(Mt|Fs) > Msa.s for all 06 s < t < ∞

A right continuous adapted process M = {Mt}t>0 is called a local martingale if thereexists a nondecreasing sequence {τk}k>1 of stopping times with τk ↑ ∞ such thatevery {Mτk∧t} is a martingale M is called a square-integrable martingale if it is amartingale and E|Mt|2

< ∞ for all t > 0 It is said to be locally square-integrable

martingale if there exists a nondecreasing sequence {τk}k>1 of the stopping timewith τk ↑ ∞ such that every {Mτk∧t} is a square-integrable martingale.

Remark 1.1.1 According to the Doob-Meyer decomposition ([2, Theorem 2.2.3]),

if Y is a submartingale process then there exists a unique predictable, increasingprocess A(t) for all t > 0 with A(0) = 0 almost surely such that the process given

by Y (t) − A(t) is a uniformly integrable martingale for each t > 0 Moreover, ifeach Y (t) = M2(t) where M is a square-integrable martingale then we can say

hM, M i(t) = A(t) for each t > 0 In this case, hM, Mi(t) is said characteristic of M.For instance, for each t > 0, if M = B(·), where B(·) is a scalar standard Brownianmotion then hM, M i(t) = t; In case M =R0tR

Uf (u) eN (ds, du), where U, f (·), eN (·, ·)can be explained in Chapter 2 of the thesis, hM, M i(t) = tR

Uf2(u)ν(du) (see e.g.[5, Remark 3.1])

To end this subsection, we list some theorems which will be frequently used inthis thesis, that are the strong law of large numbers for local martingales and theexponential martingale inequality

Lemma 1.1.1 ([52]) Let M (t), t> 0 be a local martingale vanishing at time zero.Let

ρM(t) =

Z t 0

dhM, M i(s)(1 + s)2 , t > 0

If limt→∞ρM(t) < +∞ a.s then

lim

t→∞

M (t)

t = 0, a.s.

Theorem 1.1.2 ([61, Theorem 7.4]) Let g = (g1, , gm) ∈ L2(R+; R1×m), B(t) be

an m-dimensional Brownian motion, and let T, α, β be any positive numbers Then

P

sup

06t6T

Z t 0

g(s)dB(s) − α

2

Z t 0

1.1.2 Markov process

In this subsection let us begin by saying some basic facts about Markov processes

An Ft-adapted process X = {X(t)}t>0 with values in Ξ (to be said the state space

of the process) is said a Markov process if the following Markov property is satisfied:for all 0 6 s 6 t < ∞ and A ∈ B(Ξ), P(X(t) ∈ A|Fs) = P(X(t) ∈ A|X(s)).This is equivalent to the following one: for any bounded Borel measurable function

f : Ξ → R and 0 6 s 6 t < ∞, E f (X(t))|Fs
= E(f(X(t))|X(s))

The transition probability or function of the Markov process is a function P (s, x; t, A)defined on 06 s 6 t < ∞, x ∈ Rn and A ∈ B(Ξ) with the following properties:a) For any A ∈ B(Ξ) and for every 0 6 s 6 t < ∞, P (s, X(s); t, A) = P(X(t) ∈A|X(s))

b) P (s, ·, t, A) is Borel measurable for every 0 6 s 6 t < ∞ and A ∈ B(Ξ)

c) P (s, x; t, ·) is a probability measure on B(Ξ) for every 0 6 s 6 t < ∞ and

A Markov process X is called a strong Markov process if the following strong Markovproperty is satisfied: for any bounded Borel measurable function ϕ : Ξ → R, anyfinite stopping time τ and t > 0, E ϕ(X(t + τ ))|Fτ
= E ϕ(X(t + τ))|X(τ)

If Ξ is at most countable set, any stochastic Markov process valued in Ξ is called

a Markov chain In this case, it is sufficient to consider the conditional probabilityP{X(t) = {j}|X(s) = i} for all s, t such that 0 6 s 6 t < ∞ and all i, j ∈ Ξ

If X(t) is homogenous, its transition probability depends only on t − s We oftendenote P{X(t) = {j}|X(0) = i} by Pij(t) The function Pij(t) is called standard iflim

t→0Pii(t) = 1 ∀i ∈ Ξ In this case, a state i ∈ Ξ is said to be stable if lim

t→0

1 − Pii(t)

t <

∞ Assume that Pij(t) is a standard transition function and j is a stable state Then

qij = Pij0(0) exists and is finite for all i ∈ Ξ Let qii = − lim

a right continuous step function

Furthermore, let us emphasise some notations and fundamental results on the Markovprocess

Definition 1.1.3 ([43]) A measure µ on Ξ is said invariant for Markov process

X = {Xt}t>0, if µ(A) =RΞP (t, x, A)µ(dx), ∀A ∈ B(Ξ), t > 0 An invariant measure

µ is called extremal if µ is not decomposed to the sum of two different invariantmeasures up to constant multiples

Remark 1.1.4 We can see that, a semi-group of operators {Pt}t>0 is generated

by the transition function of a Markov process Thus, for any bounded measurablefunctions f , we have Ptf (x) = R

Theorem 1.1.5 ([71, Theorem 7.4.1]) Let X = {Xt}t>0 be a Markov process, τ

be a stopping time with Ex[τ ] < ∞ and for any function twice differentiable withcontinuous second derivative f : Ξ → R with compact support in Ξ, then

Exf (τ ) = f (x) + Ex

Z τ 0

Lf (Xs)ds



This chapter considers Feller and strong Feller properties for the Markov process

X = {X(t)}t>0 For any f ∈ Bb(Ξ), the set of bounded and measurable functions,

we define Ptf (x) = Ex[f (X(t))] = E[f (Xx(t))], t > 0, x ∈ Ξ, where the family

of operators {Pt}t>0 is a semigroup of bounded linear operators on Bb(Ξ) Thesemigroup of the associated process is said to be Feller if {Pt}t>0 maps Cb(Ξ), the

set of bounded and continuous functions into itself The corresponding process issaid strong Feller if it maps Bb(Ξ) into Cb(Ξ) for each t > 0 It is known that theFeller continuity and strong Feller continuity are fundamental properties for the f -exponential ergodicity results In order to introduce the class of ergodicity properties

we have some definitions which are presented in [65] (Definitions 1.1.7-1.1.10).Definition 1.1.6 The Markov process X is called ergodic if an invariant probabilitymeasure π exists and

lim

t→∞kP (t, x, ·) − πk = 0 a.s (1.1)where the norm in (1.1) is total variation norm, that is, for any measure µ and for allmeasurable function f from (Ξ, B(Ξ)) to (Rn, B(Rn)), kµk := supR|f (x)|61f (x)µ(dx).Definition 1.1.7 The Markov process X is called uniform ergodic (strong ergodic)

if there exists an invariant probability measure π and

Definition 1.1.9 The Markov process X is called exponential ergodic if it is f exponential ergodic with at least one f > 1 such that

-kP (t, x, ·) − πk 6 M(x)ρt, ∀t > 0

If the Markov process X = {X(t)}t>0 has the Feller property, we have a theoremwhich is a key one to show the existence of a stationary distribution

Theorem 1.1.10 ([80, Theorem 2]) Let X be a Feller Markov process and let K

be a compact set Then either,

sup

x∈Ξ

1t

Z t 0

Ps1K(x)ds

→ 0 a.s., as t → ∞,

or there exists an invariant probability measure

Before introducing one of the most important properties of the Markov process,namely ergodic theory, we need to give a definition.

Definition 1.1.11 ([43]) A point x ∈ Ξ is called recurrent for Markov process

X = {X(t)}t>0 if for all open neighborhoods U of x, we have Px(RU) = 1, where

RU = {ω : X(tn)(ω) ∈ U as tn ↑ ∞} The process X is recurrent on a set A ⊂ Ξ ifall x ∈ A are recurrent

Theorem 1.1.12 ([66, Theorem 5.1]) If the Markov process X is recurrent oninvariant control set C with invariant probability µ, f (x) is a function integrablewith respect to the measure µ and all x ∈ C, then

Px lim

T →∞

1T

Z T 0

Rh X(t−), y
ν(dt, dy), with initial value X(0) = i0, in which X(t−)

is the left limit of X(t) and ν(dt, dy) is a Poison random measure with intensity

dt × µ(dy), where µ is the Lebesgue measure on R

According to the construction, we can consider a stochastic integral with respect toPoisson random measure by a continuous-time Markov chain, which is a useful tool

to improve results in the thesis

1.1.3 Lévy process

The process X = (X(t), t> 0) defined on a probability space (Ω, F, P) is said to be

a Lévy process if it satisfies the following conditions:

a) X(0) = 0 with probability 1

b) X has stationary increments, that is X(t) − X(s) has the same distribution asX(t − s), for each 0 6 s < t < ∞

c) Each X(t) − X(s) is independent of Fs, for all 0 6 s < t < ∞

d) X is stochastic continuous, i.e., for all ε > 0 and for all s> 0

lim

t→sP(|X(t) − X(s)| > ε) = 0

It is readily seen that every Lévy process has a càdlàg modification and that is itself

a Lévy process ([2, Theorem 2.1.7]) Further, standard Brownian motion, Poissonprocess, compound Poisson process, belong to the class of the Lévy process.Let ν(·) be a Borel measure defined on Ξ\{0}, the measure ν(·) is said Lévy measure

if RΞ\{0} |y|2∧ 1
ν(dy) < ∞

Remark 1.1.14 If ν(·) is a Lévy measure then ν((−ε, ε)c) < ∞, for all ε > 0.

The jump process associated to a Lévy process X is defined ∆X = (∆X(t), t > 0)where ∆X(t) = X(t) − X(t−), for each t > 0 We can see that, ∆X(t) is an adaptedprocess but it is not a Lévy process in general To count the jumps of the Lévyprocess X = {Xt}t>0, for each A ∈ B(Ξ\{0})), we define N (t, A) = #{0 6 s 6t; ∆X(s) ∈ A} = P

06s6t1A(∆X(s))

Now we brieftly explore the jumps N (·, ·) of the Lévy process In order to guaranteethat the jumps are finite, we need the following notation A set A ∈ B(Ξ\{0}) iscalled bounded below if 0 /∈ A Let S be a subset of Ξ\{0} and N (·, ·) be an integer-valued random measure on R+ × S Then N (·, ·) is said Poisson random measureif

a) For each t > 0 and A ∈ B(S) is bounded below, N (t, A) is Poisson distributed.b) If for all Ai ∈ B(S) (i = 1, , n) that are bounded below and disjoint familythen N (t1, A1), , N (tn, An) are mutually independent for each t1, t2, , tn ∈

R+

On the other hand, for each ω ∈ Ω, t > 0, the set function A → N (t, A)(ω) is

a counting measure on B(Ξ\{0}) So we define associated Borel measure ν(.) =

EN (1, ) and call it is an intensity measure associated with X Moreover, if N (·, ·) is

a Poisson random measure then its intensity measure is the product of the measureν(·) with Lebesgue measure on R+ Please see [78, pp 6] for further details

According to Doob-Meyer decomposition, we also define a compensated Poissonrandom measure by

e

N (t, A) = N (t, A) − ν(A)t (1.3)Remark 1.1.15 It should be noted that eN (t, A) is a martingale measure if ν(A) <

∞

1.2 Stochastic differential equations (SDEs)

1.2.1 SDEs with Markovian Switching

Let ξt, t > t0 (t0 > 0) be a right-continuous Markov chain on probability spacetaking in a finite state space S = {1, 2, 3, , N } with generator Γ = (qij)N ×N

−P

i6=jqij We assume that the Markov chain ξt, t > t0 is irreducible which meansthat the system will switch from any regime to any other regime Under this condi-tion, the Markov chain ξt has a unique stationary distribution π = (π1, π2, , πN) ∈

RN Furthermore, we assume that the Markov chain ξt, t > t0 is Ft-adapted but dependent of the Brownian motions B(t) = (B1(t), B2(t), , Bm(t))T, t > 0 through-out of this thesis

in-Consider a SDEs with Markovian switching

dx(t) = f (x(t), t, ξt)dt + g(x(t), t, ξt)dB(t), t0 6 t 6 T, (1.4)where coefficients f (·, ·, ·), g(·, ·, ·) of Equation 1.4 satisfy conditions for the existence

of the differential equation as in [62, Section 3.4], with initial value x(t0) = x0 and

ξt0 = ξ0, (0 6 t0 6 T ), where x0 ∈ L2

Ft0(Ω, Rn) is an Ft0-measurable Rn-valuerandom variable such that E|x0|2 < ∞ and ξ0 is an S-value Ft0 measurable randomvariable and f : Rn× R+× S → Rn

, g : Rn × R+× S → Rn×m.Now we begin to introduce theory of the stochastic differential equation with regimeswitching by presenting a theorem for existence and uniqueness of its solutions Thistheorem will play a vital role for the rest of this subsection

Theorem 1.2.1 ([62, Theorem 3.13]) Assume that there exist two positive constants

K, K such that

a) (Lipschitz condition) For all x, y ∈ Rn, t ∈ [t0, T ] and i ∈ S

|f (x, t, i) − f (y, t, i)|2∨ |g(x, t, i) − g(y, t, i)|2 6 K|x − y|2; (1.5)

b) (Linear growth condition) For all (x, t, i) ∈ Rn × [t0, T ] × S

|f (x, t, i)|2∨ |g(x, t, i)|2 6 K(1 + |x|2)

Then there exists a unique solution x(t) to equation (1.4) and, moreover

E

sup

we need a definition

Definition 1.2.2 ([59, A.1]) Let τ∞ be a stopping time which may take infinity

An Rn-valued Ft-adapted continuous stochastic process x(t), 0 6 t 6 ∞ is said alocal solution of Equation (1.4) if there is a non decreasing sequence stopping times{τk}, k = 1, 2, , N such that 0 6 τk ↑ τ∞ a.s and

x(t) = x(t0) +

Z t∧τk0

f (x(s), s, ξs)ds +

Z t∧τk0

g(x(s), s, ξs)dB(s),holds for any t> t0 and k> 1 with probability 1

If lim supt→τ∞|x(t)| = ∞ whenever τ∞ < ∞ then it is called a maximal local solutionand τ∞ is said the explosion time

Theorem 1.2.3 ([62, Theorem 3.15]) Assume that for every integer k > 1, thereexists a positive hk such that, for all t ∈ [t0, T ], i ∈ S, x, y ∈ Rn such that |x| ∨ |y| 6k,

|f (x, t, i) − f (y, t, i)|2∨ |g(x, t, i) − g(y, t, i)|2 6 hk|x − y|2.Then there exists a unique maximal local solution to equation (1.4)

Remark 1.2.4 It can be seen that, there are some different ways to prove theexistence and uniqueness of solutions (see [62, pp 89-94]) However, the locallyLipschitz condition only holds for all models in this thesis By using Theorem 1.2.3,there exists a unique maximal local solution on t ∈ [0, τe), where τe is the explosiontime In order to show this solution is global, we need to show τe = ∞, a.s We use

the technique of localization dealt with in Karatzas-Shreve [38] and Mao [61] Thistechnique will be applied in all models in this thesis From this Remark 1.2.4, wehave

Theorem 1.2.5 ([62, Theorem 3.27]) Let x(t) (t0 6 t 6 T ) be a solution of theequation (1.4) whose coefficients satisfy the conditions of the existence and unique-ness theorem Then (x(t), ξt) is a Markov process whose transition probability is given

by P (s, (x, i); t, A × {j}) = P{xx,is (t) ∈ A × {j}} for (x, i) ∈ Rn × S, A ∈ B(Rn),

j ∈ S, where xx,is (t) is the solution of equation (1.4) on t > s with initial valuex(s) = x ∈ Rn and ξs = i ∈ S

Definition 1.2.6 ([84]) Define the Markov process y(t) = (x(t), ξt) The processy(t) is said to be asymptotically stable in distribution if there exists a probabilitymeasure π(·, ·) on Rn× S such that the transition probability P (t, x, i, dy × {j}) ofy(t) converges weakly to π(dy × {j}) as t → ∞ for every (x, i) ∈ Rn× S Equation(1.4) is said to be asymptotically stable in distribution if y(t) is asymptoticallystable in distribution

The following theorem that is called the generalised Itô formula which will map thepaired process (x(t), ξt) into a new process V (x(t), t, ξt) This formula is an usefultool to consider stability of the SDEs with Markovian switching For this purpose,let C2,1(Rn × R+ × S; R) denote the family of all real-valued functions V (x, t, i)defined on Rn × R+ × S such that they are continuously twice differentiable in xand once in t Let

Theorem 1.2.7 ([62, Theorem 1.45]) If V ∈ C2,1(Rn× R+× S), then for any t > 0

V (x(t), t, ξt) =V (x(0), 0, ξ0)) +

Z t 0

LV (x(s), s, ξs)ds+

Z t 0

Vx(x(s), s, ξs)g(x(s), s, ξs)dB(s)+

Z t 0

Z

R

[V (x(s), s, i0+ (h(ξs), l)) − V (x(s), s, ξs)] µ(ds, dl),where h is defined by (1.2) and µ(ds, dl) = ν(ds, dl) − µ(dl)ds is a martingalemeasure where ν and µ have been defined in the end of subsection 1.1.2

The remainder of this subsection will present some valued theories and properties

of SDEs with Markov switching Almost of them, we refer the readers to [62] fordetails Firstly, we need a definition

Definition 1.2.8 ([62, Definition 5.34]) The trivial solution of (1.4) x(t) ≡ 0 issaid to be

• stochastically stable, if for every triple of ε ∈ (0, 1), ρ > 0 and t0 > 0, thereexists δ = δ(ε, ρ, t0) > 0 such that P {|x(t; t0, x0, i)| < ρ for all t > t0} > 1 − ε.whenever (x0, i) ∈ Bδ × S

• stochastically asymptotically stable in the large, if it is stochastically stable and,moreover: P {limt→∞x(t; t0, x0, i) = 0} = 1, for any (t0, x0, i) ∈ R+× Rn× S.Theorem 1.2.9 ([62, Theorem 5.37]) Let K be the set of continuous functions

µ : Rn+ → R+ vanishing only at 0 Moreover, function µ is said to belong to aclass of functions K∞ if µ ∈ K and µ(x) → ∞ as x → ∞ Assume that there arefunctions V ∈ C2,1(Rn× Rn× S; R+) and three functions µ1, µ2 ∈K∞ and µ3 ∈ Ksuch that

µ1(|x|) 6 V (x, t, i) 6 µ2(|x|) and LV (x, t, i) 6 −µ3(|x|) (1.6)for all (x, t, i) ∈ Rn× R+× S Then the trivial solution of Equation (1.4) is stochas-tically asymptotically stable in the large

Theorem 1.2.10 ([18, Theorem 5.1]) Assume that coefficients of the Equation(1.4) are locally Lipschitz in x for each i ∈ S and there exists a function V (·, ·) :

Rn × S → R+ that is twice continuously differentiable with respect to x ∈ Rn for

each i ∈ S and constant K > 0 satisfy

Then the Markov process (x(t), ξt) is a Feller process

Theorem 1.2.11 ([83, Theorem 2.5]) For any t ∈ [t0, T ], let

x(1)(t) = x(1)0 +

Z t 0

f1(x(1)(s), s, ξs)ds +

Z t 0

g(x(1)(s), s, ξs)dB(s), (1.7)

x(2)(t) = x(2)0 +

Z t 0

f2(x(2)(s), s, ξs)ds +

Z t 0

g(x(2)(s), s, ξs)dB(s) (1.8)Assume that coefficients of each Equation (1.7) and (1.8) satisfy the conditions

of the existence and uniqueness theorem If f1(x, t, i) > f2(x, t, i), ∀(x, t, i) ∈ R ×[t0, T ] × S and x(1)0 > x(2)0 a.s., then x(1)(t) > x(2)(t) a.s

The following notations will play a major role in the investigation a criterion toshow the existence of support of invariant probability measure Firstly, we start bytransience and recurrence notation

The Markov process (x(t), ξt) is said to be regular if and only if for any 0 < T < ∞,P{ sup

06t<T

|xx,i(t)| = ∞} = 0 This means that a process is regular if and only if itdoes not blow up in finite time

For any D ⊂ Rn, J ⊂ S and U = D × J ⊂ Rn× S, let τU := inf{t > 0 : (x(t), ξt) /∈

U }; σU := inf{t > 0 : (x(t), ξt) ∈ U } Recurrence (positive) and transience aredefined as follows Denote by σUx,i := inf{t > 0 : (xx,i(t), ξtx,i) ∈ U } A regularprocess (xx,i(t), ξtx,i) is said is recurrent with respect to U if P{σx,iU < ∞} = 1 for all(x, i) ∈ Dc× S Otherwise, the process is transient with respect to U

A recurrent process with finite mean recurrence time for some set U = D × J ,where J ⊂ S and for any D ⊂ Rn, a bounded open set with compact closure, iscalled positive recurrent with respect to U ; otherwise, the process is null recurrentwith respect to U

Theorem 1.2.12 ([86, Theorem 3.12]) If the Markov process (x(t), ξt) is recurrent(positive recurrent) with respect to some U = D × S, where D ⊂ Rn is a nonempty

open set with compact closure, then it is recurrent (positive recurrent, respectively)with respect to any bU = bD × S, where bD is any nonempty open set.

In case n = m and g is non-degenerate, i.e., g>g is positively definite, the sition probability P (t, x, i, ·) is absolutely continuous with respect the Lebesguemeasure × di and the unique invariant measure µ (if it exists) can be decomposed

tran-as µ(dx, di) = P

i∈diπiµ(dx, i), where µ(·, i) is a probability measure on Rn Letp(t, x, i, y, j) be the density of P (t, x, i, ·), due to Ichihara and Kunita [35, Proposi-tion 5.1], we have the following theorem

Theorem 1.2.13 Let the corresponding transition probabilities of the Markov cess (x(t), ξt) have an extremal invariant measure µ(dx, di) = πiϕi(x)dxdi Then

|p(t, x, i, y, j) − ϕj(y)πj|dy = 0 a.s ∀i ∈ S, x ∈ Rn

To end the subsection, we state a theorem which generalizes results of [42, Theorem4.3] In their work, jump intensity of ξt depends on the current state of the Markovchain, that is an extended case of our work

Theorem 1.2.14 ([17, Theorem 3.11]) Consider the case that Markov chain ξt isirreducible and it has a unique stationary distribution π = (π1, π2, , πN) ∈ RN Let

D be a neighborhood of 0 and a function V : D → R+which satisfies that V (x) = 0 ifand only if x = 0 and that V (x) is continuous on D, twice continuously differentiable

in D\{0} and there exists a bounded sequence of real number {ci : i ∈ S} such that

LV (x) 6 ciV (x), ∀x ∈ D\{0} If P

i∈Sciπi< 0 then the trivial solution of Equation(1.4) is asymptotically stable in probability

1.2.2 SDEs with jumps

This subsection briefly presents stochastic differential equations driven by nian motion and an independent Poisson random measure Throughout this the-sis, we always assume that, an m-dimensional standard Ft-Brown motion B(t) =(B1(t), B2(t), , Bm(t)) for each B(t) is an independent Ft-adapted Poisson randommeasure N (·, ·) defined on R+× Rn\{0} with the compensated eN (·, ·) and intensitymeasure ν(·), where ν(·) is the Lévy measure This subsection keeps definitions and

Brow-notation of [2] that is later followed by Siakalli [78].

Before presenting the SDEs, we start by introducing the Lévy-Itô stochastic integral.Let E ∈ B(Rn\{0}) and 0 6 t0 6 T < ∞ The space P2(T, E) is defined as thelinear space of all predictable, where the predictability can be seen in the sense of [2,

pp 192], mappings H : [t0, T ] × E × Ω → Rn such that RtT

For references later, we require some notations For any X = (X(t), t > t0) as

in (1.9), we define the quadratic variation process [X, X] = ([X, X](t), t > t0).The quadratic variation process is a n × n matrix-valued adapted process andits (i, j)th entry, where 1 6 i, j 6 n, is defined by [Xi, Xj](t) = [X(c)i , X(c)j ](t) +P

06s6t∆Xi(s)∆Xj(s), where X(c) is the continuous part of X defined by X(c)i =

Theorem 1.2.15 ([2, Theorem 4.4.7]) If X is a Lévy-Itô stochastic integral of theform (1.9) Then for each function f ∈ C2(Rn), t > t0, almost surely, we have