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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

I H¯C QU¨C GIA H N¸I TR×˝NG I H¯C KHOA H¯C TÜ NHI N

1.2.1 SDEs with Markovian Switching 1.2.2.

1.3 Preliminaries for stochastic mathematical models in ecosystem

Chapter 2 Long-term behavior of stochastic predator-prey systems

2.1 Dynamic behavior of a stochastic predator-prey system under regime

switching

i

2.1.1 Introduction

2.1.3

2.2 On the asymptotic behavior of a stochastic predator-prey model with

Ivlev's functional response and jumps 2.2.1 Introduction

2.2.2 Introductory results 2.2.3 Almost necessary and sucient condition for extinction and

2.2.4 Discussion and numerical example

Chapter 3 Extinction and permanence in a stochastic SIRS model

in regime switching with general incidence rate

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, AssociatePro-fessor Nguyen Thanh Dieu for his kind comments, valued suggestions, andsharing great ideas during this course

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 myparents, my wife, Cherry, for their endless care, love and patience

Hanoi, February 2020

PhD candidateTran Dinh Tuong

iii

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 withIvlev’s functional response perturbed simultaneously by the white noise and L†vynoise For the study in the epidemic model, this thesis concerns a stochastic SIRSmodel, per-turbed by both the white noise and the color noise, with a generalincidence rate We propose new approaches to provide thresholds which indicatewhether the systems are eventually extinct or permanent This allows us to derivenot only sufficient conditions but also almost necessary conditions for permanence(as well as ergodic-ity) based on sign of such thresholds Furthermore, conditionsfor the existence of stationary distributions and for the validity of the strong law oflarge numbers are established in some particular cases

Keywords Lotka-Volterra equation, predator-prey system, asymptotic behavior,ergodicity, regime switching diffusion process, stationary distribution, Ivlev’sfunc-tional response, extinction, permanence, jump diffusion process, SIRSmodel, epi-demic models

Tâm t›t

Tâm t›t Trong lu“n ¡n n y chóng tæi nghi¶n cøu d¡ng i»u ti»m c“n cıa lîp c¡c mæh…nh sinh th¡i, ÷æc mæ t£ b‹ng c¡c ph÷ìng tr…nh vi ph¥n ng¤u nhi¶n, ” mæ t£ süt÷ìng t¡c giœa thó v mçi công nh÷ nghi¶n cøu sü lan truy•n cıa dàch b»nh Łi vîi c¡ch» thó-mçi, chóng tæi nghi¶n cøu hai mæ h…nh: mæ h…nh thø nh§t vîi nhi„u tr›ng

v nhi„u m u, mæ h…nh thø hai câ ¡p øng chøc n«ng d⁄ng Ivlev bà chàu çng thíi c£nhi„u tr›ng v nhi„u L†vy Łi vîi mæ h…nh dàch b»nh, lu“n ¡n • c“p ‚n mæ h…nh t¡inhi„m SIRS bà chàu c£ nhi„u tr›ng v nhi„u m u vîi h m truy•n b»nh tŒng qu¡t.Chóng tæi • xu§t c¡c ph÷ìng ph¡p ti‚p c“n mîi ” x¥y düng c¡c gi¡ trà ng÷ïng nh‹m ch¿

ra h» ‚n mºt lóc n o â s‡ tuy»t chıng ho°c s‡ tçn t⁄i b•n vœng Düa v o d§u cıa c¡cgi¡ trà ng÷ïng, chóng tæi khæng nhœng ch¿ thu ÷æc i•u ki»n ı m cÆn r§t gƒn vîii•u ki»n cƒn cho sü tçn t⁄i b•n vœng công nh÷ t‰nh ergodic cıa h» Hìn nœa, c¡ci•u ki»n cho sü tçn t⁄i cıa c¡c ph¥n phŁi dłng công nh÷ c¡c i•u ki»n cho lu“t sŁ lîn câhi»u lüc ÷æc thi‚t l“p trong c¡c tr÷íng hæp cö th”

Tł khâa Mæ h…nh Lotka-Volterra, mæ h…nh predator-prey, d¡ng i»u ti»m c“n, t

‰nh ergodic, h» khu‚ch t¡n câ b÷îc chuy”n Markov, ph¥n phŁi dłng, ¡p øng chøcn«ng d⁄ng Ivlev, sü tuy»t chıng, sü tçn t⁄i b•n vœng, qu¡ tr…nh khu‚ch t¡n câ b÷îcnh£y, mæ h…nh SIRS, c¡c mæ h…nh dàch b»nh

v

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, whichare allowed by my coauthors to be presented in this thesis, are new and theyhave never been used in any other theses

Author

Tran Dinh Tuong

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 invariant sure settings respectively (with 1= 1) Dierent colors representdierent sizes of the density

mea-2.6 Trajectories of x(t) on the left and of y(t) on the right with intensity

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 on the

right) and

t (in red) in Ex 3.3.6

vii

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 an invariant probability measure of Ex 3.3.6 in 2D 83

3.4 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 3.3.6 in 3D 84

the indicator function of a Borel set A.

the set of natural numbers

the real line

the n-dimensional Euclidean space

f(x1; : : : ; xn) 2 Rn : x1 > 0; : : : ; xn > 0g:

f(x1; : : : ; xn) 2 Rn : x1 > 0; : : : ; xn > 0g:

Borel -algebra of S Rn:

the open ball centered at 0 with radius h > 0:

the probability space

the probability of event A

expectation of random variable X

the probability with the initial condition

the expectation corresponding with the initial condition

the minimum of a and b

the maximum of a and b

right continuous Markov chain

f1; 2; :::; Ng, the finite state space of a Markov chain

( qij)N N the generator of the Markov chain

the trace of a square matrix A = (Aij)n n,

P

x

This thesis belongs to the field of mathematical biology Although there aremany ways to think about the theory of mathematical biology, it is interesting tolook 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 tostudy the development 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]Lotka studied autocatalytic and the theory of competing species which was laterfollowed by Volterra blue by proposing the simplest Lotka-Volterra model alsoknown as the predator-prey model After that, the most general growth model,namely Kol-mogorov type, was introduced by Kolmogorov in 1936 which wasfurther developed by May, Rescigno and Richardson and Albrecht (see [76]).

On the other hand, many epidemic models that describe the dynamics of the spread ofinfectious 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.The research results are helpful to predict the developing tendency of the infectiousdisease, to decide the key elements of the spread of infectious disease and to seekthe optimum strategies of anticipating and controlling the spread of infec-tiousdiseases By computer simulations with dynamic methods, this theory could makemodeling 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 the affect ofthe disease in a population, on inoculation against smallpox Another valu-ablecontribution to the understanding of infectious diseases is a result of Simon in aoutbreak of cholera that occurred in 1854, England 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 the nextperiod of time was proportional to the number of infectious individuals multi-plied bythe number of susceptible individuals This principle was called the mass action andhas been used in many areas of science, in particular to determine the rate ofchemical reactions Now, the popular epidemic dynamic models are still so calledcompartmental 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

2

compartment (I), in which all individuals are infected by the disease and havein-fectivity; removed compartment (R), where all the individuals recovered fromthe class (I) and have permanent immunity However, some removedindividuals may lose immunity and return to the susceptible compartment Thissituation can be modeled by an SIRS model (see [32, 67, 68, 69, ]).

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 More-over,numerical experiments of specific models will be conducted to illustrate the theoreticalfindings ([64, 67, 68, ]) It has been well recognized that the traditional models areoften not adequate to describe the reality due to random environment and otherrandom factors For example, in an ecology system the growth rates and the carryingcapacities are usually changed due to environmental noise These changes usuallycannot be described by the traditional deterministic population models For instance,the growth rates of some species in the rainy season will be much different from those

in the dry season Note that the carrying capacities often vary accord-ing to thechanges in nutrition and/or food resources Likewise, the interspecific or intraspecificinteractions differ in different environments Similarly, many random factors effectepidemic Cold and flu are influenced by humidity and cold temper-atures Viruses aremore likely to survive in cold and dry conditions Hard winds, rain, cold as well as largevariations of temperatures are factors that weaken the immune system Lack ofsunlight also provokes a decrease of the level of D vitamin Clearly, the environmentand weather changes cannot be modeled as solutions of differential equations in thetraditional setup For this reason, it is more realistic to take into account the impacts ofrandom noises This demand has been stimulating the developments of stochasticmodels 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 somerandom noises In this case, stochastic differential equations (SDEs) become a usefultool to study this phenomenon with many books or papers dealing with this field (see[2, 14, 23, 31, 61, 62, 71, ] and references therein)

To make the models closer to reality, we need to consider the random perturbations ofthe environment on the population As an illustration, the distinctive seasonal changesuch as dry and rainy seasons can be affected to the population These ef-fects ofenvironment regimes in memoryless conditions to population are called color noiseand 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 ahybrid switching diffusion process in which its environment is perturbed by bothBrownian motion and Markovian switching Actually, the term hybrid sig-nifies thecoexistence of continuous dynamics and discrete events in the system The hybridswitching diffusions have a wide range application: wireless communica-tions, signalprocessing, queueing networks, production planning, biological systems, ecosystems,financial engineering, and modeling, analysis, and control and optimiza-tion of largescale systems Therefore, the systems have become a hot topic in recent years [13,

14, 17, 29, 47, ] For instance, Han and Zhao [29] studied the stability of a SIRSmodel under regime switching In 2006, Mao and Yuan [62] presented comprehensivetreatments of switching diffusion processes in this textbook The textbook coveredvarious types of stochastic differential equations with Markovian switching andemphasized the analysis of stability of the systems In [86], Yin and Zhu studied thestability of diffusion processes with state-dependent switching This textbook alsoconstructed algorithms to approximate solutions of certain systems, and providedsufficient conditions for convergence for numerical approximations to the invariantmeasures The results presented in the books [62] and [86] are useful to researchersworking in stochastic modeling, systems theory, and applications where continuousdynamics 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 explain thesephenomena, a mixture model mixing a jumps process and a diffusion process which isperturbed by white noise, are considered This model can be considered asinvolvement of the deterministic part and the random part including jumps and

environ-4

has gained much attention [2, 5, 50, 56, 91, ] We refer to the readers to [71]for providing of preliminaries of stochastic calculus with L†vy processes: Itæ-L†vydecomposition, Itæ 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 L†vy 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 conditions forthe extinction and/or persistence? A lot of works deal with these problems Forinstance, Dang, Du and Ton in [15] developed result of Rudnicki [75] to a stochasticpredator-prey system where Brownian motion acts on the coefficients of environment.They also showed that the density functions of the solutions exist and then, studiedthe asymptotic behavior of these densities In [63], Mao, Sabanis and Renshawstudied a stochastic Lotka-Volterra model by means of Lyapunov-type functions andmartingale inequalities These useful tools have been also employed extensively forestimating the growth rates, and the average in time of species in various stochasticsystems (see [24, 75, ]) Moreover, several attempts have been made to provideconditions for extinction, permanence or stability of the systems ([37, 54, .]).However, it is known that using Lyapunov function method, one can only deduce thesufficient conditions Therefore, the conditions dealt with in the above mentionedworks 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, althoughinteresting, 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

as ergodicity), they constructed a threshold parameter Based on value of thethreshold, it is known that whenever the system is permanent or some species

in the population are 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 thatthe environment is perturbed by random noises like white noise or Poisson typenoise or both Using technique in [23], we are able to derive sufficient and almostnecessary conditions for permanence (as well as ergodicity) and extinction ofthese models via threshold parameters Furthermore, convergences in totalvariation norm of transi-tion probabilities to invariant measures are expected Thefirst part is showed in the first 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-preymodel perturbed by white noise In 2011, Dang, Du and Ton [15] studied astochastic predator-prey model

pa-of system (0.1) by considering the convergence pa-of the density pa-of the solution Theprob-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

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 1 and the boundary equation has a unique stationary distribution with

6

the density f (x) Moreover, ln x(t) converges in distribution to f as t ! 1.

pro-As an extension of this work, we shall consider the model in case that the

environ-ment is perturbed by color noise

8

>

dx(t) = x(t) a ( ) b1( t)y(t)dy(t) = y(t)

The second evolution in the main first part of the thesis (Section 2.2 in Chapter2) comes from a deterministic predator-prey which was introduced by Kooij [44]

Parameters a 1 ; b 1 ; a 2 ; c 1 ; c 2 are positive real numbers, 1 e is Ivlev’s function

response In that paper, the uniqueness of limit cycles for a predator-prey system with

a functional response of Ivlev type is proved and more properties of the limit cycles

x(t)

are considered Following the method in [23], we will establish a sufficient and almostnecessary condition for permanence of a predator-prey model perturbed si-

7

multaneously by the white noise and L†vy noise with the Ivlev’s functional response

remain specific qualities will be explained in Chapter 2

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

sys-tems In the sequel, we aim to provide not only sufficient conditions but

also almost necessary conditions for permanence

Improving the estimation of convergence of x(t) to its solution on the boundary

equation (In [15], the convergence is convergence in distribution)

Establishing the conditions for the existence of the stationary distributions

and for the validity of the strong law of larger numbers for the models

Besides, in the mathematical modeling of disease transmission, SIRS epidemic

mod-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 of the thesis Because the disease transmission process is unknown in detail,

several authors proposed different forms of incidences rate in order to model the

disease transmission process under taking into account the presence of white noise,color noise and both of them For example, authors in [45] studied a deterministicSIRS model 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

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

8

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 with general 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 approach thatdoes not often work well Working on each model, the common approach is to haveglobal estimates using limit theorems of martingales and some algebraic inequalities

infec-to find conditions for extinction Besides, in order infec-to find conditions for persistence, thetechnique to introduce a Lyapunov function V and impose some conditions such thatthe Lyapunov function satisfies some desired properties such as LV < 0 outside acompact set However, this method does not provide good insight into the dynamics ofour model because the Lyapunov function is chosen subjectively rather than based onthe nature of the dynamic system Choosing dif-ferent functions gives different sets ofconditions We will treat this problem by the approach that was introduced by Du,Dang and Yin in [23] The main idea of this approach is that the persistence andextinction of a system in the interior depend strongly on the dynamics near theboundary Thus, we analyze the dynamics on the boundary, namely, stationarydistributions on the boundary and check if they are repellers or attractors Intuitively, ifall the stationary distributions on the boundary are repellers , then the system ispersistent Otherwise, extinction would happen The results obtained using theapproach are therefore very sharp and only leave critical cases unsolved As a maingoal of this part, we drive a threshold value that which determines whether the disease

is extinct or permanent Our model is the following stochastic differential equation

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+S(t)I(t)F2(S(t); I(t);

t)dB(t) ( ( t) + 1( t))R(t))dt;

(0.4)where S(t); I(t); R(t) are susceptible, infected, removed classes

contributions 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 the threshold, we obtain not only sufficientcondition but also almost necessary condition for

permanence of the disease

The thesis is organized as follows Chapter 1 presents some basicknowledge about stochastic processes as martingale process,Markov process, L†vy process and the theory of SDEs whichincludes SDEs with Markovian switching and SDEs with jumps Afterthat some notations and definitions for ecosystem and relative fieldsare given Chapter 2 and 3 are the core of the thesis Chapter 2deals with two stochastic predator-prey models In the first model, wepresent a regime switching predator-prey model perturbed by white

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

as ergodicity and extinction are investigated: when < 0, the species converge tothe population on the boundary in exponential rate If > 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 that

is used to classify the extinction and permanence of the disease In particular,

< 0 implies that the disease-free (K; 0; 0) is globally asymptotically stable,i.e., the disease will eventually disappear If > 0 the epidemic is stronglystochastically permanent Comparison to the literature and numericalsimulations to demonstrate our results are presented to end this chapter.Finally, some supplements provide additional information in Appendices sectionand Conclusions are given to end the thesis

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

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,

Ho Chi 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 tocontextualise 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, L†vy 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 preliminaries forstochastic mathematical models in ecosystem are given The basic theorems in thischapter are stated without proofs We can find proofs in the references

1.1 Stochastic processes

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

in the sets Let ( ; F; P) be a probability space A filtration is a family fFtgt>0 ofincreasing sub- -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 iscalled to satisfy the usual hypotheses if it is right continuous and F0 contains allthe P-null sets From now on, unless otherwise specified, we shall always work

on a given complete probability space ( ; F; P) with a filtration fFtgt>0 satisfyingthe usual conditions Let I be a subset of R, and be a metric space with Borel-algebra B( ) A family random variables fXtgt2I takes value in is said to be a

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stochastic process with parameter set I and state space For each fixed t 2 I,

we have a random variable Xt(!) On the other hand, for each fixed ! 2 , we have

a function t 7!Xt(!) 2 that is called a sample path (or trajectory) of the process.Throughout of this thesis, we only deal with the parameter set I = [0; 1)

Let fXtgt2R + be a stochastic process taking values in The process is said to

be c dl g (right continuous and left limit) if it is right continuous and for all mostall ! 2 the left limit lims"t Xs(!) exists and is finite for all t > 0 It is said to

be continuous (right continuous, left continuous) if for almost all ! 2 , function Xt(!)

is continuous (right continuous, left continuous) on R+ respectively It is said to be

fFtg-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; !) from R+ to is B(R+) F-measurable Let P denote the smallest -algebra on R+ with respect to which every left continuous process is a measurable function of (t; !) A stochastic

process is said to be predictable if the process regarded as a function of (t; !) is measurable Let fXtgt2R + be a stochastic process, any stochastic process fYtgt2R +

P-is said a modification of fXtg if for all t > 0; Xt = Yt a.s Here is a class of useful stochastic processes for developing some results in this thesis

1.1.1 Martingale process

A random variable : ! [0; 1] is said an fFtg-stopping time (stopping time, forthe sake of simplicity) if f! : (!) 6 tg 2 Ft for any t 2 R+ If is a stopping

time, let F = fA 2 F : A \ f! : (!) 6 tg 2 Ft; 8t > 0g that is a

sub algebra of F: A real-valued fFtg-adapted integrable process fMtgt>0 is called amartingale with respect to fFtg (or short, martingale) if EjMtj < 1 and E(MtjFs) =

Ms a.s for all 0 6 s < t < 1 We can always assume that any martingale is c dl g

A real-valued fFtg-adapted integrable process fMtgt>0 is said submartingale withrespect to fFtg (shortly submartingale) if E(MtjFs) > Ms a.s for all 0 6 s < t < 1

A right continuous adapted process M = fMtgt>0 is called a local martingale if thereexists a nondecreasing sequence f kgk>1 of stopping times with k " 1 such thatevery fM k ^tg is a martingale M is called a square-integrable martingale if it is amartingale and EjMtj2 < 1 for all t > 0 It is said to be locally square-integrable

13

martingale if there exists a nondecreasing sequence f kgk>1 of the stopping timewith k " 1 such that every fM k ^tg 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, if each Y (t)

= M2(t) where M is a square-integrable martingale then we can say hM; Mi(t) = A(t)for each t > 0 In this case, hM; Mi(t) is said characteristic of M For instance, foreach t > 0, if M = B( ), where B( ) is a scalar standard Brownian

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

If limt!1 M (t) < +1 a.s then

Theorem 1.1.2 ([61, Theorem 7.4]) Let g = (g1; : : : ; gm) 2 L2(R+; R1m), B(t) be anm-dimensional Brownian motion, and let T; ; be any positive numbers Then

supThroughout this thesis, the vast majority of the stochastic processes to be studied are Markov process

1.1.2 Markov process

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

Ft-adapted process X = fX(t)gt>0 with values in (to be said the state space

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of the process) is said a Markov process if the following Markov property issatisfied: for all 0 6 s 6 t < 1 and A 2 B( ), P(X(t) 2 AjFs) = P(X(t) 2 AjX(s)): This

is equivalent to the following one: for any bounded Borel measurable function

f : ! R and 0 6 s 6 t < 1, E f(X(t))jFs = E(f(X(t))jX(s)):

The transition probability or function of the Markov process is a function P (s; x; t; A)

defined on 0 6 s 6 t < 1, x 2 Rn and A 2 B( ) with the following properties:

a) For any A 2 B( ) and for every 0 6 s 6 t < 1, P (s; X(s); t; A) = P(X(t) 2

AjX(s)):

b) P (s; ; t; A) is Borel measurable for every 0 6 s 6 t < 1 and A 2 B( )

c) P (s; x; t; ) is a probability measure on B( ) for every 0 6 s 6 t < 1 and

R

d) The Kolmogorov-Chapman equation P (s; x; t; A) = P (u; y; t; A)P (s; x; u; dy)

holds for any 0 6 s 6 u 6 t < 1, x 2 and A 2 B( )

In terms of transition probability, the Markov property can be written as followsP(X(t) 2 AjX(s) = y) = P (s; y; t; A) for any 0 6 s < t and y 2 ; A 2 B( ) :

We say that the Markov process X is homogeneous (or has stationary transitionprobabilities) if its transition probability P (s; x; t; A) is stationary, that is, P (s +u; x; t + u; A) = P (s; x; t; A); for all 0 6 s 6 t < 1, u > 0, x 2 and A 2 B( ) In

this case, the transition probability P (s; x; t; A) depends only on t s and we can 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 strongMarkov property is satisfied: for any bounded Borel measurable function ’ : ! R, any

finite stopping time and t > 0, E ’(X(t + ))jF = E ’(X(t + ))jX( ) :

If is at most countable set, any stochastic Markov process valued in is called aMarkov chain In this case, it is sufficient to consider the conditional probabilityPfX(t) = fjgjX(s) = ig for all s; t such that 0 6 s 6 t < 1 and all i; j 2

If X(t) is homogenous, its transition probability depends only on t s We often

denote PfX(t) = fjgjX(0) = ig by Pij(t) The function Pij(t) is called standard iflim P

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

q

ij

= P 0 (0) exists and is finite for all i

ij

In this situation, the matrix

state space is finite, say S = f1; 2;

Throughout the thesis, we assume that all Markov processes are either valued

in a finite set S or valued in Rn For such a Markov chain, almost every samplepath is a right continuous step function

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

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

R

X = fXtgt>0, if (A) = P (t; x; A) (dx); 8A 2 B( ); t > 0: An invariant measure

is called extremal if is not decomposed to the sum of two different invariant measures up to constant multiples

Remark 1.1.4 We can see that, a semi-group of operators fPtgt>0 is generated bythe transition function of a Markov process Thus, for any bounded measurable

Z

Exf( ) = f(x) + Ex Lf(Xs)ds :

0

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

X = fX(t)gt>0 For any f 2 Bb( ), the set of bounded and measurable functions,

we define Ptf(x) = Ex[f(X(t))] = E[f(Xx(t))]; t > 0; x 2 ; where the family of operators

fPtgt>0 is a semigroup of bounded linear operators on Bb( ) The semigroup of theassociated process is said to be Feller if fPtgt>0 maps Cb( ), the

set of bounded and continuous functions into itself The corresponding process is said

strong Feller if it maps Bb( ) into Cb( ) for each t > 0 It is known that the Feller

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

probability measure exists and

lim kP (t; x; ) k = 0 a.s

t!1

where the norm in (1.1) is total variation norm, that is, for any measure and for all

measurable function f from ( ; B( )) to (Rn; B(Rn)), k k := sup R

R

defined for any signed measure by k kf = supjgj6f g(y) (dy), for any measurablefunction f : B( ) ! [1; 1)

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; ) k 6 M(x) t; 8t > 0:

If the Markov process X = fX(t)gt>0 has the Feller property, we have a theorem

which 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,

or there exists an invariant probability measure

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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 2 is called recurrent for Markov process

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

RU = f! : X(tn)(!) 2 U as tn " 1g The process X is recurrent on a set A if all x 2 Aare recurrent

Theorem 1.1.12 ([66, Theorem 5.1]) If the Markov process X is recurrent oninvariant control set C with invariant probability , f(x) is a function integrable withrespect to the measure and all x 2 C, then

We end this subsection by presenting a Remark which shall show relationshipbe-tween stochastic integral and a continuous-time Markov chain This propertycan be found in [61, pp 46-47]

Remark 1.1.13 It should be noted that a continuous-time Markov chain X(t)with generator = fqijgN N can be represented as a stochastic integral withrespect to a Poisson random measure Indeed, let ij be consecutive, left closed,right open intervals of the real line each having length qij such that

According to the construction, we can consider a stochastic integral with respect to Poissonrandom measure by a continuous-time Markov chain, which is a useful tool to improveresults in the thesis.

c) Each X(t) X(s) is independent of Fs; for all 0 6 s < t < 1.

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

lim P(jX(t) X(s)j > ") = 0:

t!s

It is readily seen that every L†vy process has a c dl g modification and that is itself a L†vyprocess ([2, Theorem 2.1.7]) Further, standard Brownian motion, Poisson process, compoundPoisson process, : : : belong to the class of the L†vy process

Let ( ) be a Borel measure defined on nf0g, the measure ( ) is said L†vy measure if R

nf0g

jyj2 ^ 1 (dy) < 1:

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Remark 1.1.14 If ( ) is a L†vy measure then (( "; ")c) < 1, for all " > 0.

The jump process associated to a L†vy 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 L†vy process in general To count the jumps of the L†vyprocess X = fXtgt>0, for each A 2 B( nf0g)), we define N(t; A) = #f0 6 s 6

t; X(s) 2 Ag = 06s6t 1A( X(s)):

Now we brieftly explore the jumps N( ; ) of the L†vy process In order toguarantee that the jumps are finite, we need the following notation A set A 2B( nf0g) is called bounded below if 0 2= A Let S be a subset of nf0g and N( ; )

be an integer-valued random measure on R+ S Then N( ; ) is said Poissonrandom measure if

a) For each t > 0 and A 2 B(S) is bounded below, N(t; A) is Poisson distributed

b) If for all Ai 2 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

2 R+:

On the other hand, for each ! 2 ; t > 0; the set function A ! N(t; A)(!) is a countingmeasure on B( nf0g) So we define associated Borel measure (:) = EN(1; :) andcall it is an intensity measure associated with X Moreover, if N( ; ) is a Poissonrandom measure then its intensity measure is the product of the measure

( ) with Lebesgue measure on R+ Please see [78, pp 6] for further details.According to Doob-Meyer decomposition, we also define a compensated

Poisson random measure by

Ne(t; A) = N(t; A) (A)t:

Remark 1.1.15 It should be noted that Ne(t; A) is a martingale measure if (A) < 1:

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