Sistemas dinámicos en modelos estocásticos con ruido fraccionario.

Programa de Doctorado “Matemáticas” Sistemas dinámicos en modelos estocásticos con ruido fraccionario Dynamical systems in stochastic models with fractional noise Author Javier López de la Cruz Supervisor Prof. Dr. Tomás Caraballo Garrido Co-supervisor Prof. Dr. María José Garrido Atienza A mis padres y a mi hermana A Tomás y a María José ThisworkhasbeensupportedbyJuntadeAndalucía(Andalucía,Spain)undertheProyectodeExcelencia P12-FQM-1492 and partially supported by VI Plan Propio de Investigación y Transferencia de la Universidad de Sevilla. Agradecimientos Antes de comenzar con el contenido propiamente dicho de esta tesis doctoral, me gustaría dedicar unas palabras a todas las personas que de alguna forma han formado parte del vaivén de emociones que supone realizar un proyecto como este, a los que han tenido que aguantar mis días de depresión doctoral, los que han escuchado los interesantes monólogos de quimiostatos, ecuaciones estocásticas y atractores mientras disfrutábamos de un paseo o algún refresco juntos... y, por supuesto, a aquellos con los que he celebrado los días en los que uno está eufórico porque ha conseguido, aunque sea, dar un pasito en una cuenta. Todas estas personas son también una pieza del proyecto que conforma el presente trabajo. Enprimerlugar me gustaríadar enormementelas gracias tanto ami hermana, mi Anita, comoa mis padres, Toñi y Longi, las tres personas más importantes que existen en mi vida. Gracias por todo lo que me habéis ayudadosiempre, tanto enmi vidapersonal comoen la profesional, por todo elánimo que siempre me dais, ya que sin él sería imposible conseguir cada una de las cosas que hasta ahora he conseguido, por enseñarmequedetrásdeunmatemáticotienequehaberunagranpersona...,endefinitiva, graciasporser quien sois. Hace ya diez años que salí de casa para venir a estudiar a Sevilla, aquél fue el inicio de toda esta aventura y nunca pensé que fuera a pasar tan deprisa. Parece mentira que haya pasado tanto tiempo de todas aquellas veces que me animábais a volver a Sevilla para formarme y algún día aportar algún granito de arena a la ciencia, como vosotros me decíais. Por eso, ahora que estos primeros granitos de arena han sido aportados, quiero haceros partícipes de cualquier logro que pueda conseguir. Merece una mención particular mi Anita, a la que quiero y admiro enormemente. Eres una maravillosa persona, con un corazón increíble, dispuesta siempre a ayudar a todo el que se acerca a ti y, de igual forma, al que no se acerca tanto. No he conocido persona más fuerte y que pueda enseñar tanto a todos a pesar de ser una de las más pequeñas de la familia. Este año es también un gran año para ti en tu carrera profesional, pues te conviertes oficialmente en Maestra, por tanto quiero darte mi más sincera Enhorabuena!!!! Hasdemostradoquecontrabajoypaciencialograráscualquiercosaquetepropongas,estoy seguro. Muchasgraciasportododetodocorazónsistercilla,tengoel placerdecontarconla mejorhermana que alguien pueda tener. i Resumen elpuntodevistabiológico,tambiénseremoscapacesdegarantizarlapersistenciadelasespeciesenambos casos, con y sin pared, principal objetivo desde el punto de vista biológico. Demostraremos la existencia y unicidad de solución global y también probaremos la existencia de conjuntos compactos absorbentes y atrayentes para las soluciones de los correspondientes sistemas. No obstante, el aspecto más interesante deestenuevomarcodetrabajoesquetalesconjuntosnodependerándel ruidoy,además,sedemostrarán en sentido forward, lo cual supone una diferencia significativa con respecto al resto de capítulos de esta tesis doctoral y, también, respecto a los resultados que podemos encontrar en la literatura. Finalmente, mostraremos diversas simulaciones numéricas que reflejarán los resultados demostrados a lo largo del capítulo y haremos una comparación de esta forma de modelar perturbaciones en el flujo de entrada del quimiostato con la que se usa en el Capítulo 1. Gracias a ello, podremos observar las grandes ventajas que tieneel usodelprocesodeOrnstein-Uhlenbecka la horade modelareste tipo desituaciones. Dehecho, en vista de los beneficios tan interesantes que se han logrado a lo largo del desarrollo de esta nueva idea, nos replantearemos visitar otros modelos, no necesariamente sólo los de quimiostato, ya que podrían obtenerse resultados interesantes. EnelCapítulo 3, analizamos los efectosmedioambientales causadosenlos modelosdequimiostato porel procesodeWienerestándar. Sinembargo, enestecasointroduciremoslasperturbacionesestocásticasdetalformaquetodaslasrealizacionesdelruidomantenganlassolucionesdelsistemapositivas,hecho razonable desde el punto de vista biológico. En esta ocasión, también conseguiremos ciertas mejoras respectoalosmodelosanalizadospreviamenteporotrosautores,nosóloenelcasoestocásticosinotambién eneldeterminista. Demostraremosalgunosteoremassobrelaexistenciayunicidaddesoluciónglobal,así como la existencia y unicidad de un atractor pullback aleatorio. Finalmente, mostraremos algunas simulaciones numéricas que reflejarán los resultados probados a lo largo del capítulo. Porúltimo,enelCapítulo4,analizaremoslosefectoscausadosenlosmodelosdequimiostatoporun nuevo proceso estocástico: el movimiento Browniano fraccionario, que es una generalización del proceso de Wiener o movimiento Browniano estándar. Gracias a esta nueva idea, seremos capaces de considerar diferentes tipos de términos dedifusión estoćastica que podrán ir mucho más allá de los términos aditivos o lineales multiplicativos, como ocurre el caso caso en el que usamos el proceso de Wiener estándar. Sin embargo, para poder tratar con este nuevo ruido, necesitaremos definir una nueva integral respecto del movimiento Browniano fraccionario, que llamaremos integral fraccionaria. Demostraremos la existencia y unicidad de solución global en ciertos espacios de Hölder, sin embargo, en este caso necesitaremos tratar explícitamente con el ruido con lo que necesitaremos introducir una sucesión de tiempos de parada que nos ayude a controlar el tamaño del ruido. De esta forma, probaremos también la existencia yunicidad de unatractorpullback aleatoriodiscretoy, posteriormente, estableceremos la existenciayunicidaddel correspondiente atractor pullback aleatorio continuo. Posteriormente, realizaremos un análisis de los tiempos de parada que nos permitirá dar sentido a las condiciones impuestas a lo largo del estudio matemático previo, así como realizar algunas conclusiones. Finalmente, mostraremos algunas simulaciones numéricas que nos permitirán observar los efectos de este nuevo ruido con los modelos de quimiostato. viii Abstract Thisdissertation,whichbelongstothefieldofMathematicalAnalysis andAppliedMathematics,isdedicatedtomodelandanalyzetheeffectscausedbydifferenttypesofstochasticandrandomperturbationson chemostatmodels,alaboratorydevicewhichisinterestingandusefulwhenunderstandingseveralbiologicalprocess. Thankstoconsideringstochasticprocessestoperturbthewell-knowndeterministicmodels,it is possible to obtain mathematical models which are more realistic from the biologicalpoint of view and, moreover, fitbettertherealchemostatsdisplayedinlaboratories. Inaddition,thetechniquesbasedonthe modern theory of Random Dynamical Systems as well as pullback attractors will allow us to investigate the long-time behavior of the corresponding stochastic systems and, therefore, very detailed information about the asymptotic behavior of our model will be obtained. In such a way, a decision can be made concerning which type of noise is more accurate to handle the situation to be modeled. InChapter1ouraimistoanalyzetheeffectsofstochasticdisturbancesontheinputflowinchemostat models by means of the standard Wiener process. We provide some theorems concerning the existence and uniqueness of global solution as well as the existence and uniqueness of a random pullback attractor. In addition, several numerical simulations will be shown to support the results proved through the chapter. Nevertheless,fromthepreviousstudy,somedrawbackscanbefoundwhenusingthisunbounded noisesince,forinstance,somestate variablescantakenegativevalues, whichis totallyunrealistic fromthe biological point of view even though our mathematical analysis is accurate to handle the mathematical problem. On the other hand, it is not possible to prove the persistence of the species since the disturbed inputflowcouldtakevaluesarbitrarylarge. Asaconsequence,theseinconveniencessuggestusthateither the fact of perturbing the input flow with such a noise may not be a realistic situation or that we shouldto use another different approach when modeling disturbances on the input flow of the chemostat device. In Chapter 2 we analyze the effects of random disturbances on the input flow in chemostat models but, in this case, we use a suitable Ornstein-Uhlenbeck process, involving a control parameter, which will allows us to control the noise such that every realization will remain in a strictly positive interval for every time. Thanks to this new idea, which is more realistic from the biological point of view, we can guarantee the persistence of the species in both chemostat models with and without wall growth which is, needless to say, the main goal pursued by biologists. Some theorems concerning the existence and uniqueness of ix Abstract global solution as well as the existence of compact absorbing and attracting sets for the solution of our systems are also provided. The most interesting point of this new approach it that the corresponding sets are deterministic and they are provided forwards in time, which is a significant difference with respect to the rest of the chapters on this dissertation and, of course, the results that can be found in the literature as well. Finally, several numerical simulations are shown to support the results in the chapter and compare this way of modeling disturbances on the input flow with the one used in the first chapter. In view of the previous reasons, we will notice the big advantages of making use of the corresponding suitable Ornstein- Uhlenbeckprocesswhenmodelingthiskindofsituations. Infact,inviewoftheinterestingbenefitscoming fromthis new approach, we will consider to revisit other models, not necessarily only the chemostat ones, since interesting results are expectedto be obtained. In Chapter 3, we analyze the environmental effects in chemostat models by means of the standard Wiener process. Nevertheless, differently to the analysis developed in Chapter 1, in this case we introduce stochastic disturbances in a different way, which will be properly motivated in the corresponding chapter, such that every realization of the noise keep every solution of the model in the first quadrant, which is expected from the biological point of view. In this case, we will achieve also some improvements with respect to chemostat models analyzed previously by other authors in a similar way and also with respect to the results already known about the deterministic ones. Some theorems concerning the existence and uniqueness of global solution as well as the existence and uniqueness of a random pullback attractor will be provided. Finally, some numerical simulations will be also shown to support the results through the chapter. Eventually, in Chapter 4, we analyze the effects caused on the chemostat model by a new stochastic process: the fractional Brownian motion, which is a generalization of the standard Brownian motion. Thanks to this new approach, we would be able to consider different and more general stochastic diffusion terms, not only additive or linear multiplicative ones as made when considering the standard Wiener process. Nevertheless, in order to deal with this new noise, we will need to define a new integral, called the fractional integral, respect to the fractional Brownian motion. A theorem concerning the existence and uniqueness of global solution in some damped Hölder space will be provided. However, we will need to deal with the noise explicitly such that a sequence of stopping times, which will help us to control the size ofthenoise,willbedefined. Insuchaway,wewillprovetheexistenceanduniquenessofadiscretepullback random attractor and, therefore, we will prove the existence and uniqueness of thecorresponding continuous one. Therefore, an analysis about the stopping times will be carried out in order to give sense to the conditions assumed along the previous mathematical analysis. Finally, some numerical simulations will be also shownto observe the effects of this new noise on the chemostat models. x Contents Agradecimientos i Resumen vii Abstract ix Introduction 1 Historicaloutline.......................................................................................................................... 3 Biologicalaspects......................................................................................................................... 5 Settingupthemathematicalmodel.............................................................................................. 6 Some aspects already known about the simplest deterministic model........................................... 11 Organizationandcontributionsofthedissertation ....................................................................... 13 Currentandfutureresearchlines .................................................................................................. 20 1 Modeling and analysis of stochastic disturbances on the input flow 25 1.1 Stochasticchemostatmodel................................................................................................... 26 1.1.1 Stochastic chemostat becomes a random chemostat ...................................................... 27 1.1.2 RandomchemostatgeneratesanRDS............................................................................ 27 1.1.3 Random pullback attractor for the random system ......................................................... 30 1.1.4 Random pullback attractor for the stochastic system ...................................................... 32 1.1.5 Numericalsimulationsandfinalcomments ................................................................... 34 1.2 Stochasticchemostatmodelwithwallgrowth ........................................................................ 38 1.2.1 Stochastic chemostat becomes a random chemostat ...................................................... 39 1.2.2 Numericalsimulationsandfinalcomments ................................................................... 41 2 Modeling and analysis of random disturbances on the input flow 45 2.1 A newframework to deal with the Ornstein-Uhlenbeck process............................................... 46 2.2 Randomchemostatmodel ..................................................................................................... 50 2.2.1 Existenceanduniquenessofglobalsolution................................................................... 51 2.2.2Existenceofadeterministicattractingset....................................................................... 52 2.2.3Internal structure of the deterministic attracting set ....................................................... 53 2.2.4Numericalsimulationsandfinalcomments ................................................................... 58 2.3 Randomchemostatmodelwithwallgrowth........................................................................... 64 2.3.1 Existenceanduniquenessofglobalsolution................................................................... 64 2.3.2Existenceofadeterministicattractingset....................................................................... 66 2.3.3 Internal structure of the deterministic attracting set ....................................................... 67 2.3.4Numericalsimulationsandfinalcomments ................................................................... 79 xiii 3 Modeling and analysis of environmental effects by means of the white noise 85 3.1 Stochasticchemostatmodel................................................................................................... 86 3.1.1 Stochastic chemostat becomes a random chemostat ...................................................... 88 3.1.2 Random chemostat generates a random dynamical system ............................................ 89 3.1.3 Existenceoftherandompullbackattractor..................................................................... 91 3.1.4 Random pullback attractor for the stochastic system ...................................................... 93 3.1.5 Numericalsimulationsandfinalcomments ................................................................... 94 3.2 Stochasticchemostatmodelwithwallgrowth ........................................................................ 97 3.2.1 Stochastic chemostat becomes a random chemostat ...................................................... 97 3.2.2Random chemostat generates a randomdynamical system ............................................ 98 3.2.3 Existenceoftherandompullbackattractor.....................................................................100 3.2.4Random pullback attractor for the stochastic chemostat................................................. 106 3.2.5 Numericalsimulationsandfinal comments ................................................................... 108 4 Modeling and analysis of environmental effects caused by the fractional Brownian motion 111 4.1 Chemostatmodelsandfractionalnoise .................................................................................. 112 4.2 The fractional Brownian motion andthe fractional integral..................................................... 113 4.3 Chemostat model driven by the fractional Brownian motion ................................................... 115 4.3.1 Existence and uniqueness of solution. Generation of an RDS........................................... 116 4.3.2Asymptotic dynamics of the chemostat model with fBm................................................. 118 4.3.3 Analysisofstoppingtimes.............................................................................................. 127 4.3.4Numericalsimulations................................................................................................... 132 Appendices 138 A Preliminaries on stochastic processes 139 B Preliminaries on random dynamical systems 143 C A new discrete Gronwall Lemma 147 List of Figures 151 List of Tables 153 Bibliography 155 xiv Introduction In this introductory chapter we will present the chemostat device, which will be the object of study alongthisdissertation. Someinterestinghistorical detailswill beprovidedandthemainbiologicalaspects to be taken into account will be carefully explained as well. Afterthat, we will introduce the mathematical equations describing the dynamics of the chemostat and will present some interesting ways to consider either stochasticity or randomness which will allow us to set up much more realistic models than the deterministic ones. In addition, each way will be studied in detail later in the corresponding chapter, so that adecision canbemadeconcerningthe waywhich leads intomorerealisticmodels whichfit betterthereal ones. An excellent introduction about the chemostat device, not only concerning the biological aspects but also the main mathematical ingredients necessary when beginning the study of the corresponding differentialsystems,canbefoundinthe book[49]. Duetothis fact,our introductory chapter is basedontheone providedthere,infact,someparagraphsfrom[49]willbealsomentionedinitalicswheneveritisnecessary in order to make our presentation as much nice as possible. Historical outline The chemostat is an experimental device invented in the 1950s by Monod (see [66]), and Novick and Szilard (see [67]) at the same time. On the one hand, Monod presented the equations of the chemostat as well as an experimental example in his first paper, whose aim was to control the microbial growth by interacting with the input flow. In addition, Monod proposed the name bactogène to refer the chemostat. On the other hand, Novick and Szilard presented a simpler experimental device and they found some technical difficulties when designing a system capable of supplying material to small volume reactor. Moreover, they proposed the name chemostat which comes from chemical and static. In Figure 0.1 (from Google) we cansee a photo of Jacques Monod, Aaron Novick and Leo Szilard, respectively from left to right. Figure 0.1: Jacques Monod, Aaron Novick and Leo Szilard Concerning the history of the chemostat, it was firstly used by microbiologists in order to study the growth of some species of microorganisms but its usage greatly diversified with the passage of time, in fact, such was the case that it became a prominent tool in microbiology laboratories in the 1960s, shortly after its invention, to study relationships between growth and environment parameters as well as characterizing all kinds of microbes. Nevertheless, the model as a mathematical object was seen as a formal entity. After that, in the1970s and1980s, the chemostat becamethefocus of very interestin mathematical ecologyinspiteofthefactthatitwasslightlyneglectedbymicrobiologistssincetheyweremoreinterested in the development of molecular biology approaches for the monitoring and understanding of microbial ecosystems. Infact,itwasinthe1980swhenitenjoyeditsmomentofglorywithHansenandHubell'sworks on competitive exclusion, see [47]. Later, studies on the competition of microorganisms rekindled interest among researchers, specially in the field of microbial ecology. Then, in the 2000s, when it took place the adventofthepostgenomicerawhichrequiredknowledgeandfinecontrolofreactionmedia,arenewedinterest in the chemostat device was observed among microbiologists. It was due to the fact that the model satisfies a number of formalization requirements expressed in the field of microbial ecology (see [52]), to suchanextentthatmicrobialecosystemshavetodaybecomemodelsforgeneralecology(see[54]). Finally, nowadays, the chemostat is very well understood by engineers and laboratory technicians, in such a way thatitiscurrentlyusedinscientificareasrelatedtotheacquisitionofknowledgewhichisessentialforecol- 3 willconsiderheresituationswheretheseenvironmentalvariableswillbe, ifnotoptimal,atleastconstantwithvalues somehow ideal for microorganisms. Up to now we have just mentioned the simplest deterministic chemostat model given by (3)-(4). Nevertheless, it is well known from biology that some species in the chemostat device tend to adhere to the wallsoftheculturevesselduringthebiologicalprocess. Inordertomodelthisnaturalsituationand,therefore, to analyze more realistic chemostat models, which will be called from now on chemostat model with wall growth, we also state in this section the differential equations describing its dynamics. To this end, we firstly remark that they can be obtained in a similar way to the model without taking into account the wall growth, then we just focus on the terms which are new or different from the simplest case previously presented. The deterministic chemostat model with wall growth is given bythe following differential system ds dt =D(sin −s)−ms a+sx1−ms a+sx2+bνx1,(6) dx1 dt = −(ν+D)x1+cs a+sx1−r1x1+r2x2,(7) dx2 dt = −νx2+cs a+sx2+r1x1−r2x2,(8) where s,x1,x2denoteconcentrationofthe nutrient, the microorganisms in theliquid media andtheones sticked on the walls of the culture vessel, respectively. In addition, b∈(0,1) describes the fraction of dead biomass whichis recycled, ν>0is thecollectivedeathratecoefficient ofthemicrobialbiomassrepresenting all the aforementioned factors such as diseases, aging, etc. Apart from that, 0<c≤mis the growth rate coefficient of the consumer species. Finally, r1>0and r2>0represent the rates at which the species stick on to and shear off from the walls of the culture vessel, respectively. We would like to mention here that this chemostat model with wall growth can be set up similarly to the case without taking into account the wall growth. The main difference is clearly that, in this case, we have two different species: the ones in the liquid media, denoted by x1, and the ones sticked on the walls of the culture vessel, denoted by x2. Because of this reason, we need a differential system of three equations. Concerning the equations describing the dynamics of the nutrient or susbtrate, (6), it is quite logical that we need here two terms concerning the consumption of the two different species. In addition, the term bνx1describes the microbial biomass which is recycled and hence can be considered as substrate. In (7), the term −νx1reflects the quantity of species which dies, −Dx1denotes the concentration of microbial biomass which is removed from the culture vessel to the collection vessel and, eventually, the last two termsjustreferthe quantityofmicroorganismswhichstickonto and shearoffthe wallsof the culturevessel. Finally, regarding (8), the only term which deserves to be mentioned, since the rest can be explained similarly as before, is the first one. In this case, we can observe that species are sticked on the walls of the culturevesselsotheycannotberemovedtothecollectionvesselhencewejustwrite −νx2totakeintocon- sideration the microbial biomass death as a consequence of the biological process. 10 Intherestofthedissertation,wewillanalyzebothchemostatmodels, thesimplestone(3)-(4)andthat whichtakesintoaccountthewallgrowth(6)-(8),byintroducingdifferentkindofdisturbancesineachchapter. Thiswillallowustoleadintonewandmorerealisticmodelswhichwillfitbetterthechemostatdevices displayed in laboratories in the real life. Some aspects already known about the simplest deterministic model In this section some basic results concerning the simplest deterministic chemostat model will be presented in order to compare the differences between this model (3)-(4) and the ones studied along this dissertation when considering both stochastic or random terms. It will also make us easier to realize which ways of modeling stochasticity and/or randomness are more realistic. On the other hand, the aim of this section is also to allow every reader to become comfortable with theresultsandknowledgealreadydevelopedinthe literatureaboutthesimplestdeterministicchemostat such that it is easier to realize the achievements and contributions provided in this dissertation. The following aspects are already known about system (3)-(4). We refer the interesting readers to [15, 49] for a more detailed information. ■Existence and uniqueness theorem. The theorem of existence and uniqueness of global solution can be easily proved by making use of the classical results concerning the theory of ordinary differentialequationsandthankstothefactthattheconsumptionfunctioniscontinuouslydifferentiable. ■The horizontal axis defines an invariant set. Itcanbe triviallyverified thatforany initialvalue s0for the nutrient, the mapping t7→ (s(t),x(t)) =(sin +(s0−sin)e−Dt ,0) defines a solution of system (3)-(4). ■Positivenessofsolutions. Weshouldremindthatxandsdenotequantities,orconcentrations,which are positive or equal to zero. Thus, it is essential to be sure that every solution starting with positive initial conditions remains positive or equal to zero. In order to check this property, we just have to notice that, provided s=0, we obtain ds dt =Dsin >0 whence we deduce that no trajectory can leave the positive quadrant R2 +:={(x,y) : x≥0, y≥0} since, otherwise, it should cross the horizontalsemi-axis which would contradict the uniqueness of solutions. 11 ■Another invariant set. By defining the new state variable z=s+x, it is not difficult to check that dz dt =D(sin −s)−Dx =D(sin −(s+x)) =D(sin −z) holds, whence we can obtain its solution byintegrating, which is given by z(t)=sin +((s0+x0)−sin)e−Dt for any initialpair (s0,x0)∈R2 +. Then, one can easily observe that zis approaching to sin asymptoticallywhich means that the segment I={(s,x)∈R2 +:s+x=sin} isanattractivepositivelyinvariantset. Hence,ifwerestrictthedynamicsofsystem(3)-(4)tothissegment, wecanreplace sby sin −xsuchthatequation(4)becomesthefollowingdifferentialequation dx dt =(µ(sin −x)−D)x, whose behavior is completely known as soon as the graph of the mapping x7→ γ(x)=(µ(sin −x)− D)xis known. In addition, the equality sin =s+xreflects the fact that the amount of consumed substrate is transformed into an equal amount of biomass as long as the yield is equal to one. ■Boundedness of solutions. It is automatically fulfilled by taking into account that mapping t7→ z(t)=s(t)+x(t)is bounded and the fact that both state variables sand xtake positive values or zero. ■Equilibria. It is easy to check that there exists a constant solution (sin,0), which is called washout or washed out equilibrium and corresponds to a reactor without any microbial biomass. The rest of equilibria are given by (s∗,x∗), where s∗is a value such that µ(s∗)=Dand x∗=sin −s∗hold. Intherestofthesection,wewillassumethat our growth functionµ(·)isMonodor, in other words,it is monotonicand bounded. Then, thefirst thing we wouldliketo remarkis the fact that, concerning the deterministic chemostat model (3)-(4), the term µ(s)in the equation for the substrate is a harvesting rate whereas the one appearing in the equation describing the dynamics of the species is a growthrate. Inaddition, weremarkfromthedefinitionoftheMonodfunction(5)thatthelargerthe substrate concentration is, the more significant the specific growth velocity of microorganisms is as well. By considering the Monod function as explained in the previous paragraph, we know that two different equilibria arise. On the one hand, the washout equilibrium which is given by (sin,0). On the other hand, let us define the constant λDas the unique value of the substrate concentration such thatµ(s)=Dholdstrue,aslongasD<misfulfilled.Hence,itisstraightforwardthatthegrowthvelocityof themicrobialbiomass isstrictlynegativeif s<λDwhereasitisstrictlypositivewhens>λD 12 is fulfilled. Because of this reason, the constant λDis called the break-even concentration. Let us define now s∗=λDand x∗=sin −s∗. Then (s∗,x∗)is an equilibrium with positive biomass which is globally asymptotically stable as long as D<µ(sin)holds true, which means persistence of the microbial biomass. Nevertheless, if D≥µ(sin)the washout equilibrium (sin,0) becomes globally asymptotically stable, then the species becomes extinct. Organization and contributions of the dissertation As mentioned previously, this dissertation is dedicated to model and analyze different chemostat devices affected by some stochastic and/or random perturbations. It is well-known that there exist many differentwaysofmodelingstochasticityand/orrandomnessinsomedeterministicsystem,see[5–7,42,53,74– 76,79,80]. Nevertheless, therearealsomanyquestionstobetakenintoaccountas afirststep,forinstance, we could think about the following topics: •Whichkindofstochastic/randomperturbationcanweintroduce? Therearemanydifferentstochastic processes and we need to make a decision in order to set up our stochastic/random model. Particularly, in this dissertation we will consider the standard Wiener process in the first and the third chapters,theOrnstein-UhlenbeckprocessinthesecondchapterandthefractionalBrownianmotion in the fourth one. •Howcanwedoit? Oncedecidedthestochasticprocesstoperturbthesystem,weneedtothinkwhere andhowwecanintroducethedisturbances. Forinstance,inthefirsttwochaptersofthisdissertation weperturbthedilutionrate,oritscorrespondinginputflow,motivatedbysomepersonaldiscussions with biologists and other researchers who are experts on the chemostat model whereas, in the last two chapters, we perturb the corresponding deterministic systems in a different way to model, for instance, environmental effects. We remark that each different way to perturb the original deterministic models will be properly motivatedin the corresponding chapters. •Is it realistic from the biological point of view? After thinking about the previous questions, we should think whether our resulting stochastic/random model is realistic in order to obtain models whichreflecttherealityasmuch betteraspossible.To thisend,it couldbeinteresting tomakesome numerical simulations in order to decide if the stochastic process, and of course the way in which it hasbeenintroduced, reproducewhatweexpectfromtherealityinanappropriate way. Eventhough the numerical simulations are usually made after finishing the mathematical study in order to support the results provided previously, it is also very important to try some preliminary ones when deciding the kind of disturbances which can be introduced since it helps us to obtain much more realistic systems. •Andlast butnot least... isittractablefromthemathematicalpointofview? Ofcourse,weneedour modelstobeasmuchrealisticaspossiblebutwealsoneedtohavesometractabilityinordertomake calculations,sincenoworkcouldbemadeotherwise. Sometimesonecanfoundsomemodelswhich reflect very well the real devices in a really broad sense but the model is so complicated to deal with 13 that either it is not possible to analyze its equations or just little comments can be said. Due to this facts, we need to find some suitable balance which will make our work both tractable and realistic and, then, original and interesting. After thinking carefully about these questions, we will analyze in this dissertation the effects of different types of noise and different ways of introducing stochasticity and/or randomness in the deterministic chemostat models given by (3)-(4) as well as (6)-(8). InChapter1,theeffectscausedbythestandardBrownianmotioninthechemostatmodelareanalyzed. Particularly, we are interested in introducing some disturbances on the input flow or, equivalently, in the dilutionratesuchthattheparameterDisperturbedasD+α˙ W(t),whereW(t)denotesastandardWiener processandα≥0representstheintensityofthenoise. Theresultingstochasticsystemwithouttakinginto account the wall growth is understood in It¯ o sense, is given by ds =h(sin −s)D−msx a+sidt +α(sin −s)dW (t), dx =x³ms a+s−D´dt −αxdW (t), and has already been analyzed in [77] by using the classical techniques from stochastic analysis and some stability results are provided there. Nevertheless, as in our opinion there are some unclear points in the analysis carried out there, our aim in this chapter is to use an alternative approach to this problem, specifically the theory of random dynamical systems, which will allow us to partially improve the results in [77]. In addition, we will provide some results which hold almost surely while those in [77] are just said to hold in probability. In this chapter both stochastic chemostat models with and without wall growth will be considered after introducing the disturbances explained in the previous paragraph. In both cases, a suitable variable change involving the Ornstein-Uhlenbeck process (see Appendix A for its definition) will be performed in ordertotransformouroriginalstochasticsystemintoarandomonewhichwillbemucheasiertodealwith. After that, the existence and uniqueness of global solution will be stated just like that the generation of a random dynamical system, which will be an essential ingredient henceforth. Thanks to that, the existence ofatemperedcompactrandomabsorbingsetwillbestatedwhichwillallowustoensuretheexistenceand uniqueness of a random pullback attractor associated to the random chemostat model. Moreover, we will go further by analyzing in more detail the differential equations which govern the nutrient and the microbial biomass individually, obtaining some results concerning the internal structure of the corresponding random pullback attractor. Thanks to this deeper analysis we will provide interesting results concerning the asymptotic dynamics of our system. Then, a conjugation lemma will be used to obtain the random dynamicalsystemassociatedtotheoriginalstochasticchemostatmodelaswellasitscorrespondingrandom pullback attractor. Finally, several numerical simulations will be shown to support the theoretical results proved along the whole chapter, providing also the numerical scheme used to make them, which will be considered in the following chapters as well. 14 Fromthepreviousanalysis,somedrawbackscanbefound. Ontheonehand,onecanobservethatsome state variables can take negative values which does not produce any mathematical inconsistence in our analysis or, in other words, our mathematical analysis is accurate to handle the mathematical problem. Nevertheless, from the biological point of view, this may reflect some troubles since it would mean that thereissomerevertingflowsinthepumpswhichwouldbetotallyunrealistic.Asa consequence,thisproblemsuggeststhateitherthefactofperturbingthedilutionratewithanadditivenoisemaynotbearealistic situation,orthatweshouldtrytouseanotherdifferentapproachwhenmodelingdisturbancesintheinput flow of the chemostat device. Considerations of stochastic processes in the chemostat model have already been tackled in the literature,butmainlyonthegrowthfunction(see[9,10,13]). Thisappearsparticularlyrelevantwhenthenumber ofindividualbacteriacouldbesmall,withariskofextinctionofthebiomasspopulationsinfinitetime.Nevertheless, sudden extinction in continuous cultures that are well supervised about a nominal regime are quite rare in practice. On another hand, fluctuations on the input flow that brings permanently resources to the bacterial population in continuous cultures are much likely to be observed. Hence, in Chapter 2 we willfocus onthe wayto modelthese randomfluctuations,takingintoconsiderationthatthe effective flow rate has to stay non-negative and taking also into account the wall growth in a second part. From the biological point of view, the fact of introducing a noisy term in the input flow of the chemostat models is a really interesting problem found in the laboratory since, for instance, it reflects the presence of particles of dirt inside the pumps or temporary clogs at the input or output of the chemostat. Then, it is well known that continuous flows are often subjected to random fluctuations with time. Thus,inthischapterwewillconsiderasuitableOrnstein-Uhlenbeckprocesstoperturbthedilutionrate in both chemostat models, with and without wall growth. Particularly, we are interested in perturbing D by the random term D+αz∗ β,ν(θtω), where z∗ β,ν(θtω)denotes some suitable O-U process, which will be carefully introduced in Section 2.1, and α>0represents again the amount of noise. The parameter βappearing in the O-U process will allow us to bridge a gap between a pure standard Wiener process and no noise at all. In addition, the value of such parameter is related to the amplitude of the deviations observed on the realizations. In the sequel, we will just refer to the chemostat model without wall growth in order to motivate the studycarriedout in this chaptersincethe sameargumentsarevalidforthe motivation oftheonewith wall growth, which will be also analyzed in this chapter as well. In such a way, the resulting random chemostat model without wall growth is given by the following system of differential equations ds dt =(sin −s)hD+αz∗ β,ν(θtω)i−µ(s)x, dx dt = −hD+αz∗ β,ν(θtω)ix+µ(s)x. 15 Concerning the O-U process, some essential properties will be provided in Section 2.1 which will allow us to set up a newframework and, moreover, to makecalculations in thenext sections. To sumup some of the main ingredients to be used, for every fixed event ω, it will be possible to choose βω∈Rsuch that the correspondingrealizationsof the perturbedinputflow,D+αz∗ βω,ν(θtω),will remainforevery t∈Rinside some strictly positive band which should be previously fixed, for instance, by practitioners. In such a way, for every fixed event ω, the resulting random chemostat model will be given by ds dt =(sin −s)hD+αz∗ βω,ν(θtω)i−µ(s)x,(9) dx dt = −hD+αz∗ βω,ν(θtω)ix+µ(s)x.(10) As a consequence, since βω∈Rdepends on the event ωpreviously fixed, the solutions of system (9)- (10)maynotgeneratearandomdynamicalsystem. Nevertheless,thisdoesnotrepresentanyinconvenient for the analysis of the long time behavior of the random differential system (9)-(10), since it can be investigated for every fixed event ω. In fact, we will be able to obtain some results on forwards convergence (in time) of solutions, instead of the pullback convergence ensured within the framework of random dynamical systems. Thisnewapproach,whicharisesfromthenatureoftheparticularnoise(thesuitableO-Uprocess),leads into another unusual technique which seems to be really interesting since, for instance, allows us to guarantee the existence of compact and attracting sets which are strictly positive, whence we will ensure the persistence of the species in the sense that there exists a number η>0such that, for any non null initial biomass x(0), each realization satisfies liminf t→+∞ x(t)≥η>0. (11) Needlesstosaythatthisistheprincipalgoalpursuedbybiologists,differentlytootherseveralprevious works as [53], where the authors consider disturbances in the chemostat model by means of the standard Wiener process, and prove some results concerning the persistence of the microbial biomass in the sense liminft→+∞ x(t)>0, which is clearly weaker than (11). Wewill alsoachievesomeimprovementscomparingourresultsthroughoutthischapterwiththe ones by Xu et al in [77] since, even though they consider stochastic noise on the dilution rate in the chemostat model, they need a condition on the parameter of the amplitude of the noise to ensure the persistence of the species (see, for instance, Theorem1.2 and Section 4 in [77] where the authors ensure the necessity of a smallness condition on the amount of noise α>0) whereas, in our case, modeling the disturbances with theOrnstein-Uhlenbeckprocess,thereis nodiscussionneededontheamplitudeofthenoisetoensurethe persistence,whichisinadditioninthestrongersense(11). Moreover,theauthorsin [77] saytoprovetheresultsin probability whilewewillprovealltheresultsalmostsurely,i.e., foreveryrealizationinasetofevents of full measure. 16 Apart from that, we would like to remark that some improvements are also obtained when comparing ourresults(andofcoursethewayofmodeling)withtheonesin[16]. Inthatpaper,theauthorsanalyzesome similarrandomdisturbancesontheinputflowinthechemostatmodel,withandwithoutwallgrowth,such that they replace Dby D(θtω)and, then, they prove the existence and uniqueness of a random pullback attractor by assuming that D(θtω)remains bounded inside a positive interval for all t∈R. We highlight that our analysis in this dissertation significantly differs from the one carried out in [16] since, there, the resulting random systems generate a random dynamical system and, therefore, the authors can apply the techniques involving pullback random attractors. However, we recall that our systems in Chapter 2 may notgeneratearandomdynamicalsystembut we cananalyzethelong-timebehaviorof the corresponding models without any inconvenient. In fact, this new approach allows us to prove the existence of attracting setswhicharedeterministicand,whatismore,theyareobtainedinforwardsense. Anothersignificantdifference between the work made in [16] and the one carried out in the second chapter of this dissertation is that, in our case, we know explicitly how the realizations of the perturbed dilution rate are. Thepreviousreasonsconstituteafewrepresentativeexampleswhich supportthat thiswayof perturbing the dilution rate by using the Ornstein-Uhlenbeck process fits much better the real situations we wish to model. Apart from the advantages described above, we will also obtain some improvements with respect to results obtained when analyzing the deterministic chemostat model (3)-(4), as we will explain in more detail in Chapter 2. To be more precise, in the deterministic setting the washout equilibrium (sin,0) is attractive if D=µ(sin)whereas, in our case by using the O-U process, it is possible to prove that there exists a deterministic attracting set (forwards in time) for the solutions of our system, which has several points(infact,all ofthemexceptto thewashout) insidethe positive cone. Finally, we willalsoshowseveral numerical simulations which will support the results previously proved. Thanks to that, we will be able to observeagainthe greatadvantages ofusingtheOrnstein-Uhlenbeckprocess whenmodelingstochasticity and randomness in the input flowof the chemostat model. Inviewofthat,wearereallyencouragedtothinkaboutrevisitingthepersistenceofspeciesunderinput disturbances in case, for instance, of competition between several species. In fact, as we will explained at the end of this introductory chapter, this research line is one of our main interests now. In Chapter 3, we will perturb both chemostat models, the simplest one as well as the one with wall growth, by means of the standard Brownian motion again. Nevertheless, differently to Chapter 1 where some drawbacks were found, in this chapter we will use a technique based in the one carried out by Fudenberg and Harris in [36], by Foster and Young in [34] or in [53] by Imhof and Walcher, which ensures the positivityofboththenutrientandthemicrobialbiomass,althoughdoesnotpreservethewashoutequilibrium from the deterministic to the stochastic model. More precisely, we will be interestedin analyzing the following system understood in It¯ o sense ds =h(sin −s)D−msx a+sidt −αsdW (t), (12) dx =h−Dx +msx a+sidt −αxdW (t), (13) 17 whereW(t)is a standard Brownian motion and α≥0represents the intensity ofthe noise. We remark that, in order to make the calculations more tractable and clear, we will consider the same noise in both equations, even though a similar analysis could be developed by using different Brownian motions in each equation. Now, we would like to highlight some significant insights discovered throughout this work. We will only refer to the case without wall growth since similar ones hold for the other case as well. Concerning the deterministic chemostat model (DCM) given by (3)-(4), some authors have recently proved (see [15–17]) the existence of a unique axial equilibrium (sin,0), the also called washout, which is asymptotically stable provided D>m, therefore this situation corresponds to the extinction of the microorganisms. However, if D<mand aD/(m−D)<sin the washout equilibrium becomes unstable and a unique positive globallyasymptoticallystable equilibrium appears insidethe positive quadrant, i.e., persistence of the microorganisms can be ensured. Notice that, in this case, the global attractor exists and consists of both equilibria and the heteroclinic solutions between them. Otherwise, no more information canbe deduced related to the asymptotic behavior of the system. Regarding the stochastic chemostat model (SCM) given by (12)-(13), we will prove in this chapter that there exists a unique global random attractor which is given by the singleton components (sinDρ∗(ω),0) provided D+α2/2 >m, see Section 3.1.3 in Chapter 3 for more details. Otherwise, the unique global randomattractoriscontainedin asegmentwhoseintersectionwiththe axes s=0and x=0isreducedtotwo singletonpoints. In light of the previous facts, observe that when D<mand aD/(m−D)<sin we can choose αlarge enough such that D+α2/2 >m. This means that persistence of the microorganisms holds for (DCM), while for (SCM) we have extinction since the pullback random attractor becomes the single random point (sinDρ∗(ω),0). This fact is closely related to the stabilizing effect that It¯ o's noise can produce on deterministic systems. However, if we considered a Stratonovich interpretations for our perturbation at the beginning ofour study, then we would have obtained Dinstead of D+α2/2 in the corresponding stochastic system; in other words, assumption D+α2/2 >min (SCM) would become D>m, the same that we had for(DCM).Consequently,nostabilizingeffectisproducedbythenoise(see[15,20,51]andRemark3.3in[56] for a more detailed discussion on this topic). Thus, not only the type of noise but also its mathematical interpretation can provide different results, something that has to be taken into account by the modeler. A reference that could help to make the appropriate choice in a specific application is [73], where the author presents a criterion for determining which interpretation of the noise is the most useful in his work. Finally,wewouldliketointroduceinbothdeterministicchemostatmodels,withandwithoutwallgrowth, anewstochasticprocesswhichisbecominglatelymoreandmorepopularbetweenresearchersfrommany areas: the fractional Brownian motion. Even though every detail about this new noise and its several advantages will be explained in Chapter 4, where the corresponding stochastic models will be carefully investigated, we would also like to mention in this introductory chapter some of them to motivate its study. 18 ThefractionalBrownianmotion (fBm)is acenteredGaussianprocessBHindexedbytheHurstparameter H∈(0,1). This stochastic process, which is in addition self-similar and it is characterized by the stationarity of its increments, was introduced by Kolmogorov in 1940 (see [57]) to study the long term storage capacity of reservoirs along the Nile river. This kind of noise provides us a very good candidate to model randomlong-timeinfluencesinclimatesystems, hydrologyandmedicine,betweenothers, speciallyinthe case H>1/2, as will be considered in this dissertation, since it satisfies a long-memory property then. In addition, the fBmcoincides with the standard Brownian motion when H=1/2. DifferentlytothestandardWienerprocess,thefBmisneitheramartingalenoraMarkovprocesswhen H6= 1/2. Because of that, we cannot use the It¯ o's theory to define the stochastic integral with respect to the fBm, hence it is necessary to set up a new theory to define that integral. To this end, two possibilities are available: on the one hand, we can use the so-called rough path theory which defines the corresponding integral as a point-wise limit (see [35, 64]); on the other hand, we can use the fractional calculus theory which replaces the usual ordinary derivatives by suitable fractional derivatives using its Weyl representation(see[38,43,44,65,68,72]). WewillfocusonthislattersituationwhichwillallowustodefinethestochasticintegralwithrespecttothefBmasageneralizedRiemann-Stieltjesintegral,whichiscalledthepath-wise theory. For more detailed information about this approach, we recommend every interested reader to see the pioneer works of Zähle (see [78]), Decreusefond andÜstünel (see [29]) and Lyons (see [63]). We would like to note that the modern theory of random dynamical systems, which is the base of this dissertation,stillworks eventhough the fBmandthe stochasticequationsdrivenbythemdonot generate aMarkovprocess,aswewillseeinthelastchapter. Inaddition,wewouldliketoremarkthatmanypapersin the literature treat pullback attractors for stochastic differential equations driven by a fBm with H>1/2. Nevertheless, the authors usually assume that the diffusion coefficient has a very particular form, for instance, it is additive (see [28,29]), multiplicative (see [69]) or other special cases (see [32]). In the previous cases,theseparticularformsallowtheauthorstomakeuseofacohomologymethodwhichtransformsthe original stochastic system into a random one with random parameters. However, in this dissertation we will deal directly with the stochastic chemostats driven by fBm, even though our diffusion term is linear (we notice that this work is an extension of the previous chapters where the same model, but affected by a standard Brownianmotion, was considered). Thankstothat,wewill beabletoanalyzeothermoresophisticated systems with more general diffusion coefficients with no extra work. Once introduced the new framework to deal with the fBm, in Chapter 4 the fractional Brownian motion will be presented as well as the main concepts needed to deal with the corresponding stochastic system such as the fractional derivative and the fractional integral. We will also provide some basic results concerning the integral with respect to the fBm which will be essential henceforth. In order to analyze this new stochastic system, which will be given by ds =h(sin −s)D−msx a+sidt +αsdBH(t), dx =h−Dx +msx a+sidt +αxdBH(t), 19 Stochastic disturbances on the input flow 1.1 Stochastic chemostat model Let us first recall the simplest deterministic chemostat model with Monodkinetics ds dt =(sin −s)D−msx a+s,(1.1) dx dt =x³ms a+s−D´,(1.2) where s(t)and x(t)denoteconcentrationsofthenutrient and themicrobialbiomass, respectively; sin denotes the volumetric dilution rate, ais the half-saturation constant, Dis the dilution rate and mis the maximal consumption rate of the nutrient and also the maximal specific growth rate of microorganisms. We notice that all parameters are positive and we use a function Holling type-II, which is defined as µ(s)= ms/(a+s), as functional response of the microorganism describing how the nutrient is consumed by the species (see [71] for more details and biological explanations about this model). In order to obtain a more realistic model we will perturb the dilution rate, or its corresponding input flow, in (1.1)-(1.2) by means of a white noise. Let us recall that D=qin/v, as explain in the introductory chapter, where qin denotes the input flow rate and vthe volume of the culture vessel. As vis constant in the chemostat device, it is equivalent to have disturbances on the dilution rate Dinstead of considering them on the input flow qin To this end, we will replace Dby D+α˙ W(t), where W(t)is a white noise and α≥0represents the intensity of noise. Then, system (1.1)-(1.2) becomes the following system of stochastic differential equations understood in the It¯ o sense ds =h(sin −s)D−msx a+sidt +α(sin −s)dW (t), (1.3) dx =x³ms a+s−D´dt −αxdW (t). (1.4) System (1.3)-(1.4) has been analyzed in [77] by using the classical techniques from stochastic analysis and some stability results are provided there. However, as in our opinion there are some unclear points in the analysis carried out there, our aim in this chapter is to use an alternative approach to this problem, specifically the theory of random dynamical systems, which will allow us to partially improve the results in [77]. In addition, we will provide some results which hold almost surely while those from [77] are said to hold in probability. Note that, thanks to the well-known conversion rule between It¯ o and Stratonovich formulations, we obtain from (1.3)-(1.4) its equivalent Stratonovich one given by dS =h(sin −s)¯ D−msx a+sidt +α(sin −s)◦dW (t), (1.5) dx =h−¯ Dx +msx a+sidt −αx◦dW (t), (1.6) 26 Stochastic disturbances on the input flow where ¯ D:=D+α2 2. 1.1.1 Stochastic chemostat becomes a random chemostat In this section we investigate the stochastic system (1.5)-(1.6). To this end, we first transform it into differential equations with random coefficients and without white noise by means of the following variable change which involves the stationary Ornstein-Uhlenbeck process denoted by z∗(see Appendix A). σ(t)=(s(t)−sin)eαz∗(θtω),(1.7) κ(t)=x(t)eαz∗(θtω),(1.8) whereωdenotesastandardBrownian motionandandθtωdenotestheWienershiftflow (see AppendixA for more information). For the sake of simplicity we will write z∗instead of z∗(θtω), and σand κinstead of σ(t)andκ(t). On the one hand, by differentiation, we have dσ=eαz∗(θtω)ds +α(s−sin)eαz∗(θtω)[−z∗dt +dW ] = − ¯ Dσdt −m(sin +σe−αz∗(θtω)) a+sin +σe−αz∗(θtω)κdt −αz∗σdt. On the other hand, we obtain dκ=eαz∗(θtω)dx +αxeαz∗(θtω)[−z∗dt +dW ] =m(sin +σe−αz∗(θtω)) a+sin +e−αz∗(θtω)κdt −¯ Dκdt −αz∗κdt. Thus, we have the following random system dσ dt = −(¯ D+αz∗)σ−m(sin +σe−αz∗(θtω)) a+sin +σe−αz∗(θtω)κ,(1.9) dκ dt = −(¯ D+αz∗)κ+m(sin +σe−αz∗(θtω)) a+sin +σe−αz∗(θtω)κ.(1.10) 1.1.2 Random chemostat generates an RDS Next we prove that the random chemostat given by (1.9)-(1.10) generates an RDS (see Appendix B for the definition). From now on, we will denote X:={(x,y)∈R2:x∈R,y≥0}, the upper-half plane. 27 Stochastic disturbances on the input flow Theorem 1.1.1 For any ω∈Ωand any initial value u0:=(σ0,κ0)∈X, where σ0:=σ(0;0,ω,u0)and κ0:= κ(0;0,ω,u0),system(1.9)-(1.10)possessesauniqueglobalsolutionu(·;0,ω,u0) :=(σ(·;0,ω,u0),κ(·;0,ω,u0)) ∈ C1([0,+∞),X)withu(0;0,ω,u0)=u0. Moreover,thesolutionmappinggeneratesan RDSϕu:R+×Ω×X→ Xdefined as ϕu(t,ω)u0:=u(t;0,ω,u0), for all t∈R+,u0∈X,ω∈Ω, the value at time tof the solution of system (1.9)-(1.10)with initial value u0at time zero. Proof. Observe that we can rewrite one of the terms in the previous equations as m(sin +σe−αz∗) a+sin +σe−αz∗κ=m(sin +σe−αz∗+a−a) a+sin +σe−αz∗κ=mκ−maκ a+sin +σe−αz∗ and therefore system (1.9)-(1.10) turns into dσ dt = −(¯ D+αz∗)σ−mκ+ma a+sin +σe−αz∗κ,(1.11) dκ dt = −(¯ D+αz∗)κ+mκ−ma a+sin +σe−αz∗κ.(1.12) Denoting u(·;0,ω,u0) :=(σ(·;0,ω,u0),κ(·;0,ω,u0)), system (1.11)-(1.12) can be rewritten as du dt =L(θtω)u+F(u,θtω), where L(θtω)=µ−(¯ D+αz∗)−m 0−(¯ D+αz∗)+m¶ andF:X×[0,+∞)−→ R2is given by F(ξ,θtω)=     ma a+sin +ξ1e−αz∗(θtω)ξ2 −ma a+sin +ξ1e−αz∗(θtω)ξ2     , where ξ=(ξ1,ξ2)∈X. Since z∗(θtω)is continuous, Lgenerates an evolution system on R2. Moreover, we notice that ∂ ∂ξ2·±am a+sin +ξ1e−αz∗ξ2¸= ± am a+sin +ξ1e−αz∗ 28 Stochastic disturbances on the input flow and ∂ ∂ξ1·±am a+sin +ξ1e−αz∗ξ2¸= ∓ ame−αz∗ (a+sin +ξ1e−αz∗)2ξ2, thus F(·,θtω)∈C1(X×[0,+∞);R2)which implies that it is locally Lipschitz with respect to (ξ1,ξ2)∈X. Therefore, thanks to classical results from the theory of ordinary differential equations, system (1.9)-(1.10) possessesauniquelocalsolution. Now,wearegoingto provethat theunique local solutionof system(1.9)- (1.10) is in fact a unique global one. By defining q(t) :=σ(t)+κ(t)it is easy to check that qsatisfies the differential equation dq dt = −(¯ D+αz∗)q, whose solution is given bythe following expression q(t;0,ω,q0)=q0e−¯ Dt−αRt 0z∗(θsω)ds .(1.13) The right side of (1.13) always tends to zero when tgoes to infinity since ¯ Dis positive, thus qis clearly bounded. Moreover, since dσ dt ¯¯¯¯σ=0 = − msin a+sin κ<0 wededucethat, ifthereexists some t∗>0suchthatσ(t∗)=0, wewillhaveσ(t)<0for all t>t∗. Because of the previous reasoning, we will split our analysis into two different cases. ■Case 1. σ(t)>0for all t≥0:in this case, from (1.9) we obtain dσ dt ≤ −(¯ D+αz∗)σ whose solution satisfies σ(t;0,ω,σ0)≤σ0e−¯ Dt−αRt 0z∗(θsω)ds . Since ¯ Dis positive, we deduce that σtends to zero when tgoes to infinity, hence σis bounded. ■Case 2. There exists t∗>0such that σ(t∗)=0:in this case, we already know that σ(t)<0for all t>t∗and we claim that the following bound for σholds true σ(t;0,ω,σ0)> −(a+sin)eαz∗(θtω).(1.14) 29 Stochastic disturbances on the input flow To prove (1.14), we suppose that there exists ¯ t>t∗>0such that a+sin +σ(¯ t)e−αz∗(θ¯ tω)=0, then we can findsome ε(ω)>0small enough such that σ(t)is strictly decreasing and −(¯ D+αz∗(θtω))−m(sin +σ(t)e−αz∗(θtω)) a+sin +σ(t)e−αz∗(θtω)κ(t)>0(1.15) holds for all t∈[¯ t−ε(ω), ¯ t). Hence, from (1.15) we have dσ dt (¯ t−ε(ω)) >0, thus there exists some δ(ω)>0small enough such that σ(t)is strictly increasing for all t∈[¯ t− ε(ω), ¯ t−ε(ω)+δ(ω)), which clearly contradicts the uniqueness of solution. Hence, (1.14) holds true for all t∈Rand we can also ensure that σis bounded. Sinceσ+κandσareboundedinbothcases,κisalsobounded. Hence,theuniquelocalsolutionofsystem (1.9)-(1.10) is a unique global one. Moreover, the unique global solution of system (1.9)-(1.10) remains in Xfor every initial value in Xsince κ≡0solves the same system. Finally, the mapping ϕu:R+×Ω×X→Xgivenby ϕu(t,ω)u0:=u(t;0,ω,u0), for all t≥0, u0∈X,ω∈Ω, defines an RDS. The proof of this statement follows trivially hence we omit it. 1.1.3 Random pullback attractor for the random system Now, we study the existence of the random pullback attractor, describing explicitly its internal structure. For a short description concerning the main ingredientes involved in the theory of RDSs and the notation used in the sequel, we refer the reader to Appendix B. Particularly, we would like to emphasize that E(X)denotes the set of all tempered sets, E(ω), of Xand we will take initial values in a certain tempered set E(θ−tω). Theorem 1.1.2 For any ε>0, there exists a tempered compact random absorbing set Bε(ω)∈E(X)for the RDS {ϕu(t,ω)}t≥0,ω∈Ω, that is, for any E∈E(X)and each ω∈Ω, there exists TE(ω,ε)>0such that ϕu(t,θ−tω)E(θ−tω)⊆Bε(ω), for all t≥TE(ω,ε). Proof. Thanks to (1.13), we have q(t;0,θ−tω,q0)=q0e−¯ Dt−αR0 −tz∗(θsω)ds t→+∞ −→ 0. 30 Stochastic disturbances on the input flow Then, forany ε>0thereexistsTE(ω,ε)>0such that, forall t≥TE(ω,ε)andu0∈E(θ−tω), we obtain −ε≤q(t;0,θ−tω,q0)≤ε. If we assume that σ(t)≥0for all t≥0, which corresponds to the Case 1 in the proof of Theorem 1.1.1, since κ(t)≥0for all t≥0, we have that B1 ε(ω) :={(σ,κ)∈X:σ≥0, σ+κ≤ε} is a tempered compact random absorbing set in X. Otherwise, i.e., if there exists some t∗>0such that σ(t∗)=0, which corresponds to the Case 2 in the proof of Theorem1.1.1, from (1.14), we can deduce that σ(t;0,θ−tω,u0)> −(a+sin)eαz∗(ω). Hence, we obtain that B2 ε(ω) :=n(σ,κ)∈X:−ε−(a+sin)eαz∗(ω)≤σ≤0, −ε≤σ+κ≤εo is a tempered compact random absorbing set in X. In conclusion, defining Bε(ω)=B1 ε(ω)∪B2 ε(ω)=n(σ,κ)∈X:−ε≤σ+κ≤ε,σ≥ −(a+sin)eαz∗(ω)−εo, we obtain (see Figure 1.1) that Bε(ω)is a tempered compact random absorbing set in Xfor every ε>0. σ κ −(a+sin)eαz∗(ω)−ε ε −ε B2 ε(ω) B1 ε(ω) Figure 1.1: Absorbing set Bε(ω) :=B2 ε(ω)∪B1 ε(ω) σ κ −(a+sin)eαz∗(ω) B0(ω) Figure 1.2: Absorbing set B0(ω) 31 Stochastic disturbances on the input flow Then, thanks to Proposition B.0.1 in Appendix B, it follows directly that system (1.9)-(1.10) possesses a unique random pullback attractor given by A(ω)⊆Bε(ω), for all ε>0, thus A(ω)⊆B0(ω), where B0(ω) :=n(σ,κ)∈X:σ+κ=0, σ≥ −(a+sin)eαz∗(ω)o is a tempered compact random absorbing set (see Figure 1.2) in X. Thefollowing result provides information about the internal structure of the unique random pullback attractor. Proposition 1.1.1 The unique random pullback attractor of system (1.9)-(1.10) consists of a singleton component given by A(ω)={(0,0)} as long as ¯ D>µ(sin)(1.16) holds true. Proof. We would like to note that the result in this proposition follows trivially if σremains always positive (Case 1 in the proof of Theorem 1.1.1) since in that case both σand κare positive and σ+κtends to zero when tgoes to infinity, thus the random pullback attractor is directly given by A(ω)={(0,0)}. Due to the previous reason, we only present the proof if there exists some t∗>0such that σ(t∗)=0. In such a case, we know that σ(t)<0for all t>t∗whence s(t)<sin for all t>t∗. Then, µ(s)≤µ(sin)for all t>t∗since µ(s)=ms/(a+s)is an increasing function. Hence, from (1.10) we have dκ dt ≤ −(¯ D+αz∗)κ+msin a+sin κ, which implies the following inequality κ(t;t∗,θ−tω,κ(t∗)) ≤κ(t∗)e−³¯ D−msin a+sin ´(t−t∗)−αRt∗ −tz∗(θsω)ds , where the right-hand side tends to zero when tgoes to infinity as long as (1.16) is fulfilled, therefore the unique random pullback attractor is given by A(ω)={(0,0)}. 1.1.4 Random pullback attractor for the stochastic system Wehaveprovedthatthesystem(1.9)-(1.10)hasa uniqueglobal solutionu(t;0,ω,u0)whichremainsin Xfor all u0∈Xand generates the RDS {ϕu(t,ω)}t≥0,ω∈Ω. 32 Stochastic disturbances on the input flow Now, we define a mapping T:Ω×X−→ X as follows T(ω,ζ)=T(ω,(ζ1,ζ2)) =µ(ζ1−sin)eαz∗(ω) ζ2eαz∗(ω)¶ whose inverse is given by T−1(ω,ζ)=µsin +ζ1e−αz∗(ω) ζ2e−αz∗(ω)¶. We know that v(t)=(s(t),x(t)) and u(t)=(σ(t),κ(t)) are related by (1.7)-(1.8). Since Tis a homeomorphism, thanks to Lemma B.0.1 in Appendix B we obtain a conjugated RDS which is given by ϕv(t,ω)v0:=T−1(θtω,ϕu(t,ω)T(ω,v0)) =T−1µθtω,ϕu(t,ω)µ(s(0)−sin)eαz∗(ω) x(0)eαz∗(ω)¶¶ =T−1(θtω,ϕu(t,ω)u0) =T−1(θtω,u(t;0,ω,u0)) =µsin +σ(t)e−αz∗(θtω) κ(t)e−αz∗(θtω)¶ =v(t;0,ω,v0) which means that {ϕv(t,ω)}t≥0,ω∈Ωis an RDS for our original stochastic system (1.5)-(1.6). Consequently, ϕvhas a unique random pullback attractor AT(ω)=T−1A(ω)(see Lemma B.0.2 in Appendix B) which satisfies that AT(ω)⊆BT 0(ω), where BT 0(ω) :={(s,x)∈X:s+x=sin,s≥ −a}.(1.17) Inaddition,under(1.16),theuniquerandompullbackattractorfor(1.5)-(1.6)reducestoasingletonsubset AT(ω)={(sin,0)}, which means that the microorganisms become extinct. We remark that it is not possible to provide conditions which ensure the persistence of the microbial biomass even though our numerical simulations will show that we can get it for many different values of the parameters involved in the system, as we will present in Section 1.1.5. Consequently, it is not possible to prove mathematically the persistence of the microbial biomass in our model. This drawback is owing to the fact that the standard Wiener process is a non-bounded noise, therefore the perturbed input flow D+α˙ W(t)could be occasionally large, either positive or negative. By taking into account the previous reason,itisnotsurprisingthatitisnotpossibletoensurethepersistenceofthespeciesdespiteofobtaining it numerically when the realizations of the noise are not too large such that D+α˙ W(t)remains positive 33 Stochastic disturbances on the input flow for every time. 1.1.5 Numerical simulations and final comments Tosupporttheresultsobtainedinthischapter,inthissectionwewillshowsomenumericalsimulations concerning the original stochastic chemostat model given by system(1.5)-(1.6). First of all, we will present the numerical scheme to be used since it will be the same in the rest of the dissertation. We firstly consider the following abstract system of stochastic differential equations in Stratonovich sense d X (t)=f(X(t))dt +g(X(t))◦dW (t), X(0) =X0,(1.18) where X∈Rd, and X0∈Rddenotes the value of the unique global solution of the differential equation in (1.18) at time zero. In the sequel, d=3and d=2will correspond to the stochastic chemostat models with and without wall growth, respectively. Now we define a partition ∆:={0 =τ0<τ1< ··· < τN=T}by splitting the time interval [0,T]⊂R, T>0,into Nsubintervals andsettingδt=T/Nandτj=jδt, for j=0,...,N. Ouraimistoapproximate X(τj)≈Xjby using the Euler-Maruyama method (see [50] for more detailed information). In this way we integrate the differential equation in (1.18) on τj−1≤t≤τjfor some arbitrary j∈ {0,...,N}and we use the following approximations of both deterministic and stochastic integrals Zτj τj−1 f(X(s))ds ≈f(Xj−1)δt and Zτj τj−1 g(X(s))dW (s)≈g(Xj−1)δWj, where δWj:=W(τj)−W(τj−1)∼N(0,δt)are independent normally distributedrandom variables. Hence, we can already define the following numerical scheme given by Xj=Xj−1+F(Xj−1)δt+G(Xj−1)δWj for every j=1,...,N. Concerning our stochastic chemostat model, it can be then discretized as follows sj=sj−1+F1(xj−1,sj−1)∆t+G1(xj−1,sj−1) (W(τj)−W(τj−1)), xj=xj−1+F2(xj−1,sj−1)∆t+G2(xj−1,sj−1) (W(τj)−W(τj−1)), 34 Stochastic disturbances on the input flow for every j=1,...,N, where F1,G1,F2,G2:Rd−→ Rare functions given by F1(xj−1,sj−1)=(sin −sj−1)¯ D−msj−1xj−1 a+sj−1 , F2(xj−1,sj−1)=xj−1µmsj−1 a+sj−1 −¯ D¶, G1(xj−1,sj−1)=α(sin −sj−1), G2(xj−1,sj−1)=αxj−1. Having reached this point, we present different numerical simulations to show the dynamics of our original stochastic chemostat model (1.5)-(1.6). Particularly, we display the phase plane (s,x)of the correspondingmodel. Thebluedashedlinesrepresentthesolutionsofthedeterministic(i.e.,withα=0)system (1.1)-(1.2) and the other ones are different realizations of the stochastic chemostat model (1.5)-(1.6). In addition, we set sin =1,a=0.6,m=3and we consider (s(0),x(0)) =(2.5,5) as initial pair. We also present different cases where the value of the dilution rate and the amount of noise change in order to obtain different situations in which condition (1.16) is (or is not) fulfilled. On the one hand, in Figure 1.3 we take D=3and we choose α=0.1 (left) and α=0.5 (right). In both cases, it is easy to check that ¯ D=1.5050 (left), ¯ D=1.6250 (right) and µ(sin)=1.8750, thus thanks to Proposition 1.1.1, we knowthat the microorganisms become extinct, as we show in the simulations. s(t) 0 0.5 1 1.5 2 2.5 x(t) 0 1 2 3 4 5 6Phase plane s(t) -0.5 0 0.5 1 1.5 2 2.5 3 x(t) 0 1 2 3 4 5 6 7 8Phase plane Figure 1.3: Extintion. α=0.1 (left) and α=0.5 (right) On the other hand, in Figure 1.4we take D=3but, in this case, α=1(left) and α=1.5 (right). Then, it followsthat ¯ D=2(left)and ¯ D=2.6250 (right)then,sinceµ(sin)=1.8750 andthankstoProposition1.1.1, we also obtain the extinction of the species. 35 Stochastic disturbances on the input flow On the one hand, in Figure 1.8 we take D=3,b=2,ν=0.2,c=1.5 and we choose α=0.5. We can see that every state variable remains in the first octant in this case, i.e., they are all positive, however both species, the one in the medium or liquid media and also the one sticked on the walls, become extinct. Figure 1.8: Extinction of both species. α=0.5 Similarly, in Figure 1.9 we take D=3,b=2,ν=0.2,c=1.5 and we increase the intensity of the noise to α=1.5. In this case, both species become extinct and, moreover, the substrate reaches negative values for some times which is totallyunrealistic from the biological point of view, as explained previously. 42 Stochastic disturbances on the input flow Figure 1.9: Extinction of both species. α=1.5 Now, in Figure 1.10 we take D=3,b=0.5,ν=1.2 and α=0.5. In this case, it is not difficult to notice that both species persist. Nevertheless, the dynamics of the substrate clearly cross the line s=0which is an important inconvenient. Figure 1.10: Persistence of both species. α=0.5 43 Stochastic disturbances on the input flow Finally, in Figure 1.11 we take D=3,b=0.5,ν=1.2 and α=1.5. In this case, the species which are sticked on the walls of the culture vessel persist whereas the ones in the liquid media become extinct. In addition, we also find some drawbacks since the substrate reaches negative values. Figure 1.11: Persistence and extinction. α=1.5 44 C h a p t e r 2 Modeling and analysis of random disturbances on the input flow in chemostat models In this chapter, we will consider both chemostat models, with and without wall growth, where the input flow is perturbed by means of the Ornstein-Uhlenbeck process which is, differently to the standard Wiener process, a boundednoise. In this case, we will set up a new framework. Particularly, a parameter β willbeintroducedasdriftintheLangevinequationthatallowstobridgeagapbetweenapureWienerprocessandnonoise atall. This parameter,whosevalueis relatedto theamplitudeof thedeviationsobserved onthe realizations,actsaa controlinpractice. Insucha way, theresulting randomdifferentialsystemmay not generate a random dynamical system, nevertheless this does not represent any inconvenient for the analysis of its long-time behavior, since it can be investigated for every fixed event ω. Thanks to this new approach, we will prove the existence and uniqueness of positive global solution of our models as well as the existence of both deterministic absorbing and attracting sets for the solutions of our system, in order to obtain detailed information regarding its asymptotic dynamics. However, the most interesting point is that such sets will be deterministic and will be obtained forwards in time, which is a relevant achievement respect to the case when dealing with random dynamical systems and pullback attractors. In addition, we will ensure the previous sets to be strictly positive, which means that the persistence of the species will be ensured. Thatis,needlesstosay,themainpurposeofbiologists. Finally,severalnumericalsimulationswill be presented to support the provided results. Apart from that, some comparisons between the way of perturbing the input flow in this chapter and Chapter 1 will be stated and some discussions and conclusions concerning the use of bounded noise to perturb the input flow in the chemostat models will be also provided. The results and explanations concerning the contributions of this chapter can be found in [12,60]. 45 Random disturbances on the input flow 2.1 A new framework to deal with the Ornstein-Uhlenbeck process Inthisfirstsection,wesetanewframeworkwhichwillbehighlyusefulandwillconsistinthekeytoensure the persistence of the species under some smallness conditions on certain parameters involved in the chemostat model such as the dilution rate, the amplitude or intensity of the noise and also the collective death rate of the species in the model with wall growth. This new approach, in addition, will allow us to prove the existence of both absorbing and attracting sets forwards in time and, what is more interesting, independently on the noise. At first,letus considerWbeinga twosidedstandardWienerprocess. Asexplainedin AppendixA,Kolmogorov's theorem ensures that Whas a continuous version, that we will denote by ω, whose canonical interpretationisasfollows: letΩ=C0(R,R), thespaceofcontinuousfunctionsthatarezeroatzero,Fthe Borel σ−algebra on Ωgenerated by the compact open topology (see [4] for details) and Pthe corresponding Wiener measure on F. We consider now the Wiener shift flow given by θtω(·)=ω(·+ t)−ω(t), for all t∈R. Then, (Ω,F,P,{θt}t∈R)defines a metric dynamical system (see Appendix A for details). Now, we introduce a new suitable Ornstein-Uhlenbeck process, as a generalization of the one defined in (A.1) in Appendix A, by considering some parameters on the drifts of the stochastic process which allow us to control the realizations of the noise in some sense. To this end, we present the following Ornstein- Uhlenbeckprocessonthemetricdynamicalsystem(Ω,F,P,{θt}t∈R),whichisdefinedastherandomvariable given by z∗ β,ν(θtω)= −βν 0 Z −∞ eβsθtω(s)ds,for all t∈R,ω∈Ω,β>0, ν>0, (2.1) which solves the following Langevin equation (see [4,18,19]) dz +βzdt =νdω(t), for all t∈R.(2.2) The Ornstein-Uhlenbeck process given by (2.1) is a stationary mean-reverting Gaussian stochastic process where β>0is the mean reversion constant that represents the strength with which the process is attracted by the mean or, in other words, how strongly our system reacts under some perturbation, and ν>0is the volatility constant which represents the variation or the size of the noise independently of the amount of the noise α>0. In fact, the Ornstein-Uhlenbeck process can describe the position of some particlebytakingintoaccountthefriction,whichisthemaindifferencewiththestandardWienerprocessand makesourmodeltobeabetterapproachto therealones. Inaddition,theOrnstein-Uhlenbeckprocesscan be understood as a kind of generalization of the standard Wiener process, which would correspond to take β=0andν=1in (2.1). 46 Random disturbances on the input flow By taking into account the definition of both parameters βand νinvolved in the Langevin equation (2.2), we highlight the following relevant observations concerning the effects caused by each of them on the evolution of the stochastic process. •Fixed β>0.Then, the volatility of the process is larger if we consider a larger ν. However, the evolution of the process is considerably smoother when we take a smaller value of ν. This is quite reasonable since νdecides the amount of noise introduced to dz, which measures the variation of the stochastic process, hence the process will be subjected to suffer much more changes when choosing a larger value of ν. WecaneasilyobservethisbehaviorinFigure2.1wherewesimulatetwotypicalrealizationsoftheperturbed dilutionrateD+αz∗ β,ν(θtω)withD=2,α=0.8,β=2andweconsiderν=0.1(blue)andν=0.5 (orange). Figure 2.1: Realizations of the perturbed dilution rate with D=2,α=0.8 andβ=2 •Fixed ν>0.Then, the process tends to go further away from the mean value if we consider a smaller valueofβ. However,theattractionofthemeanincreaseswhentakingalargerβ,whichisabsolutelylogical since βhas a huge influence on the drift of the Langevin equation (2.2). For instance, we can observe this behavior in Figure 2.2, where we simulate two realizations of the perturbed dilution rate D+αz∗ β,ν(θtω) with D=2,α=0.8,ν=0.5 and we take β=2(blue) and β=10 (orange). 47 Random disturbances on the input flow Figure 2.2: Realizations of the perturbed dilution rate with D=2,α=0.8 andν=0.5 WeestablishbelowanimportantresultinvolvingsomeergodicpropertiesheldbytheOrnstein-Uhlenbeck process (2.1) which will be used at several places in the sequel. Proposition 2.1.1 There exists a θt-invariant set e Ω∈Fof Ωof full P−measure such that for ω∈e Ωand β,ν>0, we have (i) the random variable |z∗ β,ν(ω)|is tempered (see Definition B.0.3in Appendix B). (ii) the mapping (t,ω)→z∗ β,ν(θtω)= −βν 0 Z −∞ eβs(θtω)(s)ds is a stationary solution of (2.2)with continuous trajectories; (iii) for any ω∈˜ Ωone has: lim t→±∞ |z∗ β,ν(θtω)| t=0; lim t→±∞ 1 tZt 0 z∗ β,ν(θsω)ds =0; lim t→±∞ 1 tZt 0¯¯¯z∗ β,ν(θsω)¯¯¯ds =Ehz∗ β,νi< ∞; (iv) finally, for any ω∈e Ω, lim β→+∞ z∗ β,ν(θtω)=0, for all t∈R. Remark 2.1.1 Wenotethat theproof of(iv)can befoundin [1] (see Lemma4.1)and wereferthe readersto[4,18] for the proof of (i)-(iii). 48 Random disturbances on the input flow Then, we restrict the metric dynamical system previously introduced to e Ω, such that we obtain a new metric dynamical system, see [14] for details. For simplicity, we denote again this new metric dynamical system bythe old symbols, namely (Ω,F,P,{θt}t∈R). Our aim in this section is to analyze both chemostat models with and without wall growth, where the input flow is perturbed by using the O-U process as explained before. To this end, let us first fix a strictly positive interval, namely (b1,b2)⊂R, where b2>b1>0. Thanks to the last item in Proposition 2.1.1, for each ω∈Ω, it is possible to choose β∈Rlarge enough such that the corresponding realization of the perturbed input flow, D+αz∗ β,ν(θtω), remains inside the interval (b1,b2)for every t∈R. Nevertheless, it is not possible to ensure, from a theoretical point of view, that there exists some β∈Rsuch that almost all realizations of the perturbed input flow remains in (b1,b2), even though it seems to be true when making some numerical simulations. Because of this reason, we will analyze our random chemostat models, with and without wall growth, for every fixed ω∈Ω. In the sequel, let us refer only to the chemostat model without wall growth in order to motivate our analysis, sincethe same arguments are valid formotivating the case with wall growth. As stated above, we know that it is possible to find βω∈Rsuch that D+αz∗ βω(θtω)∈(b1,b2)for every t∈R, then we need to analyze the following random system ds dt =(sin −s)hD+αz∗ βω,ν(θtω)i−µ(s)x,(2.3) dx dt = −hD+αz∗ βω,ν(θtω)ix+µ(s)x.(2.4) We would like to remark that the choice of βdepends on ω∈Ω, this is the reason to write βωin the previous system. Then,βωactsinpracticeasacontrolparameter. Asaconsequence, sinceβω∈Rdepends on the event ωpreviously fixed, the solutions of system (2.3)-(2.4) may not generate a random dynamical system. Nevertheless, this does not represent any inconvenient for the analysis of the long-time behavior of the random differential system (2.3)-(2.4) since it can be investigated, as pointed out before, for every fixed event ω. Therefore, once fixed an event ω∈Ω, we have that βωis also a fixed real number, thus we willrewriteβω=β,forthesakeofclarity. Theinterestingfactisthattheattractingsetsforthesolutionswill notdependonω,sowewillobtainanonrandomsetwhereallsolutionsforallrealizationswillapproachto. Throughout the rest of the section, we prove the existence and uniqueness of a global solution of the corresponding system (2.3)-(2.4) aswell as the existence of a strictly positive forward attracting set for the solutions of system (2.3)-(2.4), under some condition involving the parameters of the model, whence we willbeabletoensurethatthemicroorganismconcentrationwillbealsoasymptoticallyinsideastrictlypositive interval or, in other words, we will guarantee the persistence of the species. As pointed out previously, the same arguments are used to motivate the analysis of the corresponding random chemostat model with wall growth, as we will see in Section 2.3. 49 Random disturbances on the input flow Before starting with the analysis previously motivated, let us define the following constants s:=µ−1(b1)and ¯ s:=µ−1(b2), (2.5) which will be essential henceforth. In addition, we recall that µ(s)=ms a+s,for alls≥0, denotes the consumption function of the species. In Figure 2.3 where we plot the mapping s7→ µ(s)and overlap a realization of the perturbed input flow as well, without taking into account the dependency of time. D b2 b1 s¯ s m s Figure 2.3: Realizations of the perturbed dilution rate, sand ¯ s Once presented this new framework, we will focus in the next sections on providing a careful analysis of both random chemostat models, with and without wall growth. 2.2 Random chemostat model Inthissectionweanalyzethesimplestchemostatmodelwithoutwallgrowthwheretheinputflowhas been perturbed by using the Ornstein-Uhlenbeck process, which will lead us to deal with a random differential system in place of the stochastic one studied in Chapter 1. Specifically, we are interested in analyzing the following random chemostat model ds dt = −hD+αz∗ β,ν(θtω)is−µ(s)x+sin hD+αz∗ β,ν(θtω)i,(2.6) dx dt = −hD+αz∗ β,ν(θtω)ix+µ(s)x.(2.7) 50 Random disturbances on the input flow Henceforth,ω∈Ωisfixedandβ∈RisalsoaparameterwhichhasbeenfixedsuchthatD+αz∗ β,ν(θtω)∈ (b1,b2)for all t∈R. 2.2.1 Existence and uniqueness of global solution In this section, X={(x,y)∈R2:x≥0, y≥0} denotes the positive cone. Theorem 2.2.1 For any initial pair v0:=(s0,x0)∈X, system (2.6)-(2.7)possesses a unique global solution v(·;0,ω,v0) :=(s(·;0,ω,v0),x(·;0,ω,v0)) ∈C1([0,+∞),X) with v(0;0,ω,v0)=v0, where s0:=s(0;0,ω,v0)and x0:=x(0;0,ω,v0). Proof. We set v(·;0,ω,v0) :=(s(·;0,ω,v0),x(·;0,ω,v0)) such that system(2.6)-(2.7) can be rewritten as dv dt =L(θtω)v+F(v,θtω), where L(θtω)=Ã−(D+αz∗ β,ν(θtω)) −m 0−(D+αz∗ β,ν(θtω))+m! andF:X×[0,+∞)−→ R2is given by F(ξ,θtω)=   ma a+ξ1 ξ2+sinD+αsin z∗ β,ν(θtω) −ma a+ξ1 ξ2  , where ξ=(ξ1,ξ2)∈X. Sincez∗ β,ν(θtω)iscontinuous,LgeneratesanevolutionsystemonR2. Moreover,wenoticethatF(·,θtω)∈ C1(X×[0,+∞);R2)which implies that it is locally Lipschitz with respect to (ξ1,ξ2)∈X. Therefore, system (2.6)-(2.7) possesses a unique local solution. Now,weprovethattheuniquelocalsolutionofsystem(2.6)-(2.7)isdefinedforanyforwardtimeandis, then, aunique globalone. Tothisend, wedefinethe newstatevariable q(t) :=s(t)+x(t)−sin. Therefore, qsatisfies the following differential equation dq dt = −hD+αz∗ β,ν(θtω)iq,(2.8) whose solution is given by q(t;0,ω,q0)=q0e−Dt−αRt 0z∗ β,ν(θsω)ds .(2.9) 51 Random disturbances on the input flow 2.2.4 Numerical simulations and final comments In this section we would like to show some numerical simulations concerning the random chemostat model (2.6)-(2.7) in order to support the results proved throughout this section. We remark that we use again the Euler-Maruyama method (see [50] for more detailed information) to define the numerical scheme necessary to obtain the numerical simulations, see Section 1.1.5 in Chapter 1 or [9,10,13]. In every simulation, the dashed lines represent the solution of the deterministic systems, i.e., the behavior of the stochastic/random system after taking α=0, whereas the continuous lines correspond to different realizations of the solution of the corresponding stochastic/random system. Now, we show some simulations concerning the random chemostat model previously studied in Section2.2. Ineachofthefollowingfiguresthreepanelsaredisplayed: theleftoneshowsthephaseplaneand thegeneraldynamicsoftherandomchemostatmodel;thetwopanelsontheright-handsidehelpustosee two important zones in the phase plane. These places are said to be important due to the fact that some notable changes can be observe in the dynamics of the system around these points. InFigure2.5wesetD=2,sin =4,a=0.6,m=5,α=0.5,β=1,ν=0.7andinitialvaluess0=2,x0=5 for the nutrient and the microorganisms, respectively. In this case (2.14) holds true and this is the reason whywecanobservethepersistenceofthespecies. Wecanalsoseehowtherealizationsareapproachingto the line s+x=sin, as proved in (2.12) and (2.20). s(t) 0 0.5 1 1.5 2 x(t) 3 3.5 4 4.5 5 5.5 6Phase plane s(t) 0 0.5 x(t) 5 5.2 5.4 5.6 5.8 6 6.2 Upper-Left Zoom s(t) 0 0.5 x(t) 5 5.2 5.4 5.6 5.8 6 6.2U-L Extra Zoom s(t) 0 0.5 x(t) 3.4 3.6 3.8 4 4.2 Lower-Left Zoom s(t) 0 0.5 x(t) 3.4 3.6 3.8 4 4.2 L-L Extra Zoom Figure 2.5: Persistence of the species in the random chemostat model 58 Random disturbances on the input flow In the next two figures, we change the intensity of the noise andthe parameters involved in the definitionoftheOrnstein-Uhlenbeckprocess. Wewilljustspecifythoseparameterstobechangedinordernotto beredundant. Forinstance,inFigure2.6weincreasetheintensityofthenoisetoα=1,themeanreversion constanttoβ=5andweslightlydecreasethevolatilityconstanttoν=0.2. Inthiscasewecaneasilynotice that the dynamics tends to the line s+x=sin, as previously proved. s(t) 0 0.5 1 1.5 2 x(t) 3.5 4 4.5 5 5.5 6Phase plane s(t) 0.2 0.4 0.6 x(t) 5.4 5.5 5.6 5.7 5.8 5.9 Upper-Left Zoom s(t) 0.2 0.4 x(t) 5.6 5.65 5.7 5.75 5.8 5.85 U-L Extra Zoom s(t) 0.2 0.4 x(t) 3.5 3.6 3.7 3.8 3.9 Lower-Left Zoom s(t) 0.4 0.5 x(t) 3.55 3.6 3.65 3.7 3.75 3.8 L-L Extra Zoom Figure 2.6: Persistence of the species in the random chemostat model (ii) Furthermore, in Figure 2.7 we take α=4and the rest of parameters are not changed respect to the last ones. We can also see the dynamics approaching to x+s=sin even though the quantity of noise is considerablylarge. Infact,bytakingintoaccountthemodelperturbedbyusingthewhitenoise,asmadein Chapter1,wecancheckthehighdifferencesbetweenbothwaysofmodelingthedisturbancesontheinput flow since here we can consider large values as α=4whereas much smaller values, such as α=1or even α=0.5, made some state variables take negative values in the case of considering the standard Wiener process. It is basically due to the fact that, in this case where we use the Ornstein-Uhlenbeck process, the parameterβplaysanessentialroleinthesensethattheintensityofthenoisecanbeincreasedinexchange for increasing β. 59 Random disturbances on the input flow s(t) 0 0.5 1 1.5 x(t) 3.5 4 4.5 5 5.5 6 Phase plane s(t) 0 0.20.4 0.60.8 x(t) 4.5 5 5.5 6 Upper-Left Zoom s(t) 0.2 0.4 0.6 x(t) 5.2 5.4 5.6 5.8 6 U-L Extra Zoom s(t) 0 0.20.4 0.60.8 x(t) 3.4 3.6 3.8 4 4.2 Lower-Left Zoom s(t) 0.2 0.4 0.6 x(t) 3.4 3.6 3.8 4 4.2 L-L Extra Zoom Figure 2.7: Persistence of the species in the random chemostat model (iii) In Figure 2.8 we take D=3.5,sin =2,a=0.8,m=0.5,α=0.5,β=1,ν=0.7 and initial values s0=2.5,x0=5for the nutrient and the microorganisms, respectively. Then we can see that the microorganisms extinguish, what is not surprising due to the fact that condition (2.13) is fulfilled. We can also see here how the realizations are approaching the line s+x=sin, as proved in (2.12). s(t) 1.8 2 2.2 2.4 2.6 x(t) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Phase plane s(t) 1.9 1.95 2 x(t) 1.34 1.36 1.38 1.4 1.42 1.44 1.46 Upper-Left Zoom s(t) 1.92 1.94 1.96 x(t) 1.48 1.49 1.5 1.51 1.52 1.53 1.54 U-L Extra Zoom s(t) 1.61.8 2 2.22.4 x(t) 0 0.2 0.4 0.6 0.8 Lower-Left Zoom s(t) 1.98 1.99 2 x(t) 0.01 0.02 0.03 0.04 0.05 0.06 L-L Extra Zoom Figure 2.8: Extinction of the species in the random chemostat model As made before, we only specify the parameters to be changedrespect to the last ones considered. For example, in Figure 2.9 we increase the intensity of the noise and the mean reversion constant to α=1and 60 Random disturbances on the input flow β=5, respectively. In addition, we slightly decrease the volatility constant to ν=0.2. In this case, it is straightforward to notice that, since (2.13) is true, the dynamics tend to (sin,0), as proved in Proposition 2.2.1. s(t) 1.8 2 2.2 2.4 2.6 x(t) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Phase plane s(t) 1.9261.928 1.93 x(t) 1.367 1.3675 1.368 1.3685 1.369 1.3695 1.37 1.3705 Upper-Left Zoom s(t) 1.9351.9361.937 x(t) 1.502 1.5025 1.503 1.5035 1.504 1.5045 1.505 1.5055 U-L Extra Zoom s(t) 1.98 2 2.02 x(t) 0 0.01 0.02 0.03 0.04 0.05 0.06 Lower-Left Zoom s(t) 1.98 2 2.02 x(t) 0 0.02 0.04 0.06 0.08 0.1 0.12 L-L Extra Zoom Figure 2.9: Extinction of the species in the random chemostat model (ii) On the other hand, we change in Figure 2.10 the intensity of the noise to α=4and we observe the samebehavior,eventhoughthequantityofnoiseisconsiderablylargerespecttotherestofthecaseswhen perturbing the input flow by using the Ornstein-Uhlenbeck process and, what is more interesting, respect to the case when using the standard Brownian motion, as in Chapter 1. s(t) 1.8 2 2.2 2.4 2.6 x(t) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Phase plane s(t) 1.9 1.95 x(t) 1.35 1.36 1.37 1.38 1.39 1.4 1.41 Upper-Left Zoom s(t) 1.92 1.93 1.94 x(t) 1.49 1.495 1.5 1.505 1.51 1.515 1.52 U-L Extra Zoom s(t) 1.9 2 2.1 x(t) 0 0.05 0.1 0.15 0.2 Lower-Left Zoom s(t) 1.95 2 2.05 x(t) 0 0.05 0.1 0.15 0.2 L-L Extra Zoom Figure 2.10: Extinction of the species in the random chemostat model (iii) 61 Random disturbances on the input flow Next we present two figures where we overlap a typical realization of the solution of system (1.5)-(1.6) and another one of the solution of system (2.6)-(2.7) such that we can notice much more easily the differences between the simulations concerning the different ways of modeling randomness and stochasticity on the input flow. In each figure we display a big panel where the general dynamics can be seen and four smaller panels which correspond to two different zooms of two interesting places of the realizations, specifically the dynamics around (s,x)=(2,2) and (s,x)=(2,0) in Figure 2.11 and the dynamics about (s,x)=(0.2,5.5) and(s,x)=(0.4,3.75) in Figure 2.12. In Figure 2.11 we plot a typical realization when perturbing the dilution rate with the standard Wiener process (orange) and two different ones when perturbing with the Ornstein-Uhlenbeck process for different values of the mean reversion constant β=2(red) and β=0.5 (green). In this case, we take sin =2, D=3.5,a=0.8,m=0.5,α=0.8,σ=0.8,x0=5and s0=2.5. We can easily observe that (2.13) and (1.16)arebothfulfilledthenthemicroorganismsbecomeextinct, aswealreadyprovedinSections1.1.3 (see Proposition (1.1.1)) in Chapter 1 and 2.2.3 (see Proposition (2.2.1)) in Chapter 2. s(t) 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 x(t) 0 1 2 3 4 5 6Phase plane s(t) 1.9 1.95 2 2.05 2.1 x(t) 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 Upper-Left Zoom s(t) 1.97 1.98 1.99 2 2.01 x(t) 1.9 1.95 2 2.05 2.1 U-L Extra Zoom s(t) 1.92 1.94 1.96 1.98 2 x(t) 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Lower-Left Zoom S(t) 1.95 1.96 1.97 1.98 1.99 x(t) 0.05 0.1 0.15 0.2 0.25 L-L Extra Zoom Figure 2.11: Comparison in case of extinction Eventually, in Figure 2.12 we plot again a typical realization when perturbing the dilution rate with the 62 Random disturbances on the input flow standardWienerprocess(orange)andtwodifferentoneswhenperturbingbyusingtheOrnstein-Uhlenbeck process for β=2(red) and β=0.5 (green), but now we take sin =4,D=2,a=0.6,m=5,α=0.15, σ=0.8,x0=5and s0=2. In this case (1.16) does not hold true, thus it is not possible to ensure the persistence of the species (in the chemostat model perturbed by using the standard Wiener process) although numerically it can be obtained for the previous values of the parameters. In addition, D<µ(sin) and ¯ s<sin hold true, then we can ensure the persistence of the microbial biomass when perturbing the chemostat model by means of the Ornstein-Uhlenbeck process. Moreover, we can observe that every realization is approaching to the line s+x=sin, as proved in Section 1.1.4 (see (1.17)) in Chapter 1 and Section 2.2.3 (see (2.20)) in Chapter 2. s(t) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x(t) 3 3.5 4 4.5 5 5.5 6Phase plane s(t) 0.1 0.2 0.3 0.4 x(t) 5.3 5.4 5.5 5.6 5.7 5.8 Upper-Left Zoom s(t) 0.2 0.25 0.3 0.35 x(t) 5.6 5.65 5.7 5.75 5.8 5.85 U-L Extra Zoom s(t) 0.2 0.3 0.4 0.5 x(t) 3.55 3.6 3.65 3.7 3.75 3.8 3.85 3.9 3.95 4 Lower-Left Zoom s(t) 0.3 0.35 0.4 0.45 x(t) 3.6 3.65 3.7 3.75 3.8 L-L Extra Zoom Figure 2.12: Comparison in case of persistence In conclusion, we can observe that the Ornstein-Uhlenbeck process gives us a useful and interesting tool whenmodeling stochasticityandrandomnesssinceitallowsustosetupmathematicalmodels which guarantee the positiveness of the variable and therefore it better suits to represent reality. Moreover, the absorbingandattractingsetsprovidedinthemathematicalresultsaredeterministicandforwardsintime. This new framework could also allow us to revisit the persistence of species under input disturbances, for instance, in case of taking also intoaccount the wall growth, as we will develop in Section 2.3. 63 Random disturbances on the input flow 2.3 Random chemostat model with wall growth In this section, we are interested in analyzing the chemostat model with wall growth (6)-(8) where the input flow is perturbed by means of the Ornstein-Uhlenbeck process, similarly to Section 2.2. 2.3.1 Existence and uniqueness of global solution We are interested in analyzing the following random differential system with wall growth and Monod kinetics ds dt =³D+αz∗ β,ν(θtω)´(sin −s)−ms a+sx1−ms a+sx2+bνx1,(2.22) dx1 dt = −³ν+D+αz∗ β,ν(θtω)´x1+cs a+sx1−r1x1+r2x2,(2.23) dx2 dt = −νx2+cs a+sx2+r1x1−r2x2,(2.24) where z∗ β,ν(θtω)denotes, as in Section 2.2, the new Ornstein-Uhlenbeck process defined in (2.1). Henceforth, we consider ω∈Ωto be fixed and, then, β∈Ris also a fixed parameter such that D+αz∗ β,ν(θtω)∈ (b1,b2)for every t∈R. In this section, X={(x,y,z)∈R3:x≥0, y≥0,z≥0} will denote the positive cone in the threedimensional space. Firstly, we will state a result concerning the existence and uniqueness of globalsolutionof the chemostat model with wall growth (2.22)-(2.24). Theorem 2.3.1 For any initial triple v0:=(s0,x10,x20)∈X, system (2.22)-(2.24)possesses a unique global solution v(·;0,ω,v0) :=(s(·;0,ω,v0),x1(·;0,ω,v0),x2(·;0,ω,v0)) ∈C1([0,+∞),X) with v(0;0,ω,v0)=v0, where s0:=s(0;0,ω,v0),x10 :=x1(0;0,ω,v0)and x20 :=x2(0;0,ω,v0). Proof. Let us recall that the random system (2.22)-(2.24) can be rewritten as dv dt =L(θtω)v+F(v,θtω), where L(θtω)=    −³D+αz∗ β,ν(θtω)´−m+bν−m 0−³ν+D+αz∗ β,ν(θtω)´−r1+c r2 0r1−ν+c−r2     64 Random disturbances on the input flow andF:X×[0,+∞)−→ R3is given by F(η,θtω)=          ³D+αz∗ β,ν(θtω)´sin +ma a+η1 η2+ma a+η1 η3 −ca a+η1 η2 −ca a+η1 η3          , where η=(η1,η2,η3)∈X. Sincez∗ β,ν(θtω)iscontinuous,LgeneratesanevolutionsystemonR3. Moreover,wenoticethatF(·,θtω) ∈C1¡X×[0,+∞);R3¢whichimpliesthatitislocallyLipschitzwithrespectto(η1,η2,η3)∈X. Therefore, system (2.22)-(2.24) possesses a unique local solution. Now,weprovethattheuniquelocalsolutionofsystem(2.22)-(2.24)isdefinedforanyforwardtimeand is, then, a unique global one. To this end, we define the new state variable p(t)=s(t)+m c(x1(t)+x2(t)) and take into account that D+αz∗ β,ν(θtω)>b1>0for every t∈R,c≤mand b≤1. Thus we have that p satisfies the next random differential inequalities dp dt =³D+αz∗ β,ν(θtω)´sin −³D+αz∗ β,ν(θtω)´s−hm c³ν+D+αz∗ β,ν(θtω)´−bνix1−m cνx2 ≤³D+αz∗ β,ν(θtω)´sin −b1s−m cb1x1−m cνx2 ≤³D+αz∗ β,ν(θtω)´sin −ϑhs+m cx1+m cx2i =³D+αz∗ β,ν(θtω)´sin −ϑp(t), or, in other words, pverifies the following random differential equation dp dt ≤³D+αz∗ β,ν(θtω)´sin −ϑp(t), (2.25) where ϑ:=min{b1,ν}>0. By solving (2.25), we have p(t;0,ω,p0)≤p0e−ϑt+sin Zt 0³D+αz∗ β,ν(θsω)´e−ϑ(t−s)ds.(2.26) Weremarkthat theintegrandin(2.26) converges to zerofor every t≥s≥0when tgoes to infinity, but not the integral. Moreover, the integral has subexponential growth. 65 Random disturbances on the input flow Therefore, pdoes not blow up at any finite time, thus s,x1and x2do not blow up at any finite time either. Hence,thesolutionof system(2.22)-(2.24)is definedforanyforwardtime,whencewe canstraightforwardly deduce that the unique local solution of our system (2.22)-(2.24) is, in fact, a unique global one. Now, wearegoingtoprovethatthepreviousuniqueglobal solutionremainsinthepositiveconeXfor every initial value v0∈X. To this end, we firstly consider x1≥0and x2≥0and we evaluate the random differential equation for the substrate when s=0such that we have ds dt ¯¯¯¯s=0 =³D+αz∗ β,ν(θtω)´sin +bνx1>0 due to the fact that the perturbed input flow is always positive. Moreover, for every s≥0and x2≥0, from the equation of the microorganisms in the liquid media, we have dx1 dt ¯¯¯¯x1=0 =r2x2≥0 and, for every s≥0and x1≥0, from the species which are sticked on the walls of the culture vessel, we have dx2 dt ¯¯¯¯x2=0 =r1x1≥0. Thus, the unique global solution v(t;0,ω,v0)of our random system (2.22)-(2.24) remains in the positive cone Xfor every initial value v0∈X. 2.3.2 Existence of a deterministic attracting set Inthis section, we study the existence ofadeterministiccompactabsorbing set as wellastheexistence of adeterministic attracting set, both of themforwardsin time, for thesolutionsofour randomchemostat model with wall growth (2.22)-(2.24). Theorem 2.3.2 For any given ε>0, there exists a deterministic compact absorbing set Bε⊂Xfor the solutions of system (2.22)-(2.24), i.e., there exists TF(ω,ε)>0such that for every given initial pair v0∈F, the solution corresponding to v0remains inside Bεfor all t≥TF(ω,ε). Proof. Consider again the state variable p(t)=s(t)+m c(x1(t)+x2(t)). Then, from(2.26) we obtain p(t;0,ω,p0)≤p0e−ϑt+sin Zt 0³D+αz∗ β,ν(θsω)´e−ϑ(t−s)ds ≤p0e−ϑt+sin Zt 0 b2e−ϑ(t−s)ds 66 Random disturbances on the input flow =p0e−ϑt+sinb2 ϑh1−e−ϑti,(2.27) since D+αz∗ β,ν(θsω)≤b2for every s∈R. As a consequence, after making tgo to infinity in (2.27), we have lim t→+∞ p(t;0,ω,p0)≤sinb2 ϑ.(2.28) From (2.28) we know that, for every initial value p0∈Fand any given ε>0, there exists some time TF(ω,ε)>0such that 0≤p(t;0,ω,p0)≤sinb2 ϑ+ε for all t≥TF(ω,ε). Thus, Bε=½(s,x1,x2)∈X:s+m c(x1+x2)≤sinb2 ϑ+ε¾ is,foranyε>0,adeterministiccompactabsorbingset(forwardsintime)forthesolutionsofsystem(2.22)- (2.24). Therefore, thanks to Theorem 2.3.2, it can be easily deduced that B0:=½(s,x1,x2)∈X:s+m c(x1+x2)≤sinb2 ϑ¾(2.29) isadeterministicattractingset(forwardsintime)forthesolutionsofthechemostatmodelwithwallgrowth (2.22)-(2.24). 2.3.3 Internal structure of the deterministic attracting set In this section, our aim is to analyze the internal structure of the deterministic attracting set B0, given by(2.29). Tothisend,weperformtheusualvariablechangewhenstudyingthechemostatmodelwithwall growth by defining two new state variables, the total biomass and the proportion of the microorganisms in the medium, respectively, as follows x(t)=x1(t)+x2(t)and ξ(t)=x1(t) x1(t)+x2(t).(2.30) For the sake of simplicity, we will write xand ξinsteadof x(t)andξ(t). 67 Random disturbances on the input flow +cs(t;0,ω,s0) a+s(t;0,ω,s0)x(t;0,ω,x0). (2.51) On the one hand, thanks to the definition of the proportion (2.30), we have that 0≤ξ(t;0,ω,ξ0)≤1(2.52) for every t≥0and any initial value ξ0∈(0,1). Thus,from(2.51),thankstothepreviouscalculationsandRemark2.3.1,weobtainthefollowingrandom differential inequality dx(t;0,ω,x0) dt ≥ −νx(t;0,ω,x0)−b2x(t;0,ω,x0) +cs(t;0,ω,s0) a+s(t;0,ω,s0)x(t;0,ω,x0)(2.53) for all t≥0and every initial value v0∈F. By definition, we know that z(t;0,ω,z0)=cs(t;0,ω,s0)+mx(t;0,ω,x0) and,thanksto (2.46),we havethat, foreach v0∈Fandany ε>0, thereexists sometime TF(ω,ε)>0such that z∗ l−ε≤cs(t;0,ω,s0)+mx(t;0,ω,x0)≤z∗ u+ε holds true for every t≥TF(ω,ε). As a consequence, since c≤m, we have the following inequalities cs(t;0,ω,s0)≥z∗ l−ε−mx(t;0,ω,x0) and s(t;0,ω,s0)≤z∗ u c+ε c−x(t;0,ω,x0) for every initial value v0∈F, anyε>0and for all t≥TF(ω,ε). Then, from (2.53), we have dx(t;0,ω,x0) dt ≥ −νx(t;0,ω,x0)−b2x(t;0,ω,x0) +z∗ l−mx(t;0,ω,x0)−ε a+z∗ u c+ε c−x(t;0,ω,x0) x(t;0,ω,x0)(2.54) 74 Random disturbances on the input flow for every v0∈F, anyε>0and for all t≥TF(ω,ε). Now, we study the differential equation (2.54) when x=ex, where exis defined as ex= z∗ l−(ν+b2)³a+z∗ u c´ m+c.(2.55) Then, from (2.54) and considering ε<cex, we obtain dx(t;0,ω,x0) dt ¯¯¯¯x=ex ≥"−(ν+b2)+z∗ l−mex−ε a+z∗ u c+ε c−ex#ex >"−(ν+b2)+z∗ l−mex−cex a+z∗ u c+cex c−ex#ex=0 for every v0∈F, anyε∈(0,cex)and for all t≥TF(ω,ε). Hence, as long as (2.50) is fulfilled, we have that, for any ε∈(0,cex), where exis given by (2.55), for every v0∈F, there exists some time TF(ω,ε)>0such that dx(t;0,ω,x0) dt ¯¯¯¯x=ex >0(2.56) for all t≥TF(ω,ε). Therefore,from(2.56)weconcludethat, as long as (2.50)isfulfilled,we havethefollowinglowerdeterministic bound for the dynamics of the species x(t;0,ω,x0)>ex, for any ε∈(0,cex), every given v0∈Fand for all t≥TF(ω,ε). Now, let us recall the random differential equation held by the substrate ds dt =³D+αz∗ β,ν(θtω)´(sin −s)−ms a+sx+bνξx for every t≥0, whence we obtain that ds(t;0,ω,s0) dt =³D+αz∗ β,ν(θtω)´sin −³D+αz∗ β,ν(θtω)´s(t;0,ω,s0) −ms(t;0,ω,s0) a+s(t;0,ω,s0)x(t;0,ω,x0) 75 Random disturbances on the input flow +bνξ(t;0,ω,ξ0)x(t;0,ω,x0), (2.57) for all t≥0and every initial value v0∈F. Moreover, from (2.46) and (2.52), we know that, for each v0∈Fand any ε>0, there exists some time TF(ω,ε)>0such that x(t;0,ω,x0)≤z∗ u m+ε m holds true for every t≥TF(ω,ε). Summing up, thanks to the previous calculations, from (2.57) we have ds(t;0,ω,s0) dt ≥b1sin −b2s(t;0,ω,s0) −ms(t;0,ω,s0) a+s(t;0,ω,s0) z∗ u+ε m,(2.58) for every v0∈F, anyε>0and for all t≥TF(ω,ε). Now, we study the differential equation (2.58) when s=es, where esis defined as es=b1sin b2+2z∗ u a .(2.59) Then, from (2.58) and considering ε<z∗ u, we obtain ds(t;0,ω,s0) dt ¯¯¯¯s=es ≥b1sin −b2es−es a+es(z∗ u+ε) >b1sin −b2es−2es az∗ u=0 for every v0∈F, anyε∈(0,z∗ u)and for all t≥TF(ω,ε). Hence, we have that, for any ε∈(0,z∗ u)andevery v0∈F, there exists some time TF(ω,ε)>0such that ds(t;0,ω,s0) dt ¯¯¯¯s=es >0 for all t≥TF(ω,ε). Thus, we obtain the following lower deterministic bound for the dynamics of the substrate s(t;0,ω,s0)>es, 76 Random disturbances on the input flow for any ε∈(0,z∗ u), every given v0∈Fand for all t≥TF(ω,ε). In conclusion, we obtain that, by taking any ε∈(0,min{cex,z∗ u}), where exis given by (2.55), for every given v0∈F, there exists some time TF(ω,ε)>0such that x(t;0,ω,x0)>ex(2.60) and s(t;0,ω,s0)>es(2.61) hold true for all t≥TF(ω,ε). As a result, we can deduce that, for anyε∈(0,min{cex,z∗ u}), b B(s,x) ε=©(s,x)∈R2 +:x≥ex,s≥es,z∗ l−ε≤cs +mx ≤z∗ u+εª,(2.62) where exand esare defined by (2.55) and (2.59), respectively, is a deterministic compact absorbing set (forwards in time) for the solutions of system (2.31)-(2.32). It is worth mentioning that we already proved the dynamics of system (2.31)-(2.32) to remain inside B(s,x) ε, defined as in (2.47), forwards in time. In the previous result, as long as (2.50) is fulfilled, we obtain in addition a smaller deterministic compact absorbing set e B(s,x) εforwards in time as well, defined by (2.62), which besides is strictly contained in the first quadrant of the two-dimensional space. This fact will bethemainkey,asalreadyhappenedinthecasewithoutwallgrowth,toguaranteethepersistenceofboth species, individually, in the strong sense (11). s x ex es z∗ u m z∗ l m z∗ l c z∗ u c Figure 2.14: Attracting set e B(s,x) 0 77 Random disturbances on the input flow Therefore, e B(s,x) 0:=©(s,x)∈R2 +:x≥ex,s≥es,z∗ l≤cs +mx ≤z∗ uª is a deterministic attracting set (forwards in time) for the solutions of system (2.31)-(2.32), see Figure 2.14. Remark 2.3.2 It is not difficult to check that both ex<z∗ l mand es<z∗ l care satisfied. Finally, wewillanalyzethe dynamicsofboth species, x1and x2,individually,toprovethatbothofthem also persist as long as (2.50) holds true. To this end, thanks to , and the definition of the proportion, ξ= x1/x, we obtain that x1(t;0,ω,x10)=ξ(t;0,ω,ξ0)x(t;0,ω,x0) >ξ∗ lex>0 for every tlarge enough and any initial value v0∈F. In addition, we also have x2(t;0,ω,x20)=x(t;0,ω,x0)(1−ξ(t;0,ω,ξ0)) >(1−ξ∗ u)ex>0 for every tlarge enough and any initial value v0∈F. Hence, we obtain that both species, the ones in the medium and also the ones sticked on to the walls of the culture vessel, will persist as long as (2.50) holds true. Remark 2.3.3 It is possible to improve the deterministic lower bounds obtained in (2.60) and (2.61) by considering smaller values of ε>0. Particularly, we could consider ε∈¡0, c nex¢, for any n∈N, instead of ε∈(0,cex)such that we get that x>exn, instead of x>exas in (2.60), where exnis given by exn:= z∗ l−(ν+b2)³a+z∗ u c´ m+c n , which clearly satisfies exn:= z∗ l−(ν+b2)³a+z∗ u c´ m+c n > z∗ l−(ν+b2)³a+z∗ u c´ m+c=:ex>0, for any n∈N, since n≥1. 78 Random disturbances on the input flow Similarly, we could consider ε∈¡0, 1 nz∗ u¢, for any n∈N, instead of ε∈(0,z∗ u)such that we get s>esninstead of s>esas in (2.61), where esnis given by esn:=b1sin b2+z∗ u a¡1+1 n¢ which clearly verifies esn:=b1sin b2+z∗ u a¡1+1 n¢>b1sin b2+2z∗ u a =:es>0, for any n∈N, since n≥1. Summing up, in order to ensure the persistence of both species, the ones in the medium and the ones which are sticked on the walls of the culture vessel, the conditions to be imposed need to be the following ones ν<c−Dξ∗ land ν+b2<z∗ l a+z∗ u c ,(2.63) where ξ∗ l,z∗ land z∗ uare deterministic constants definedby (2.38) and (2.45), respectively. We would like to highlight that both conditions in (2.63) essentially represent some restrictions on the dilution rate, on the disturbances on the input flow and also on the death collective rate, which is totally logical from the biological point of view. Inparticular,ifthedilutionrate, oritsequivalentinputflow, weretoolarge,thenthemicrobialbiomass would not be able to have access to the nutrient which would mean the extinction of both species and, furthermore, much more quantity of microbial biomass would be removed from the culture vessel to the collection vessel which would also increase significantly the probability of the extinction. In addition, the disturbancesontheinputflowcannotbetoolargesincewewanttoavoidthedrawbacksfoundwhenmodeling the disturbances by means of the white noise. This is also the main reason which encouraged us to think about another kind of way to perturb the input flow in the chemostat model, particularly by using the Ornstein-Uhlenbeck process. Apart from that, it made also us think about introducing this kind of stochastic processes when modeling other situations,such as the one with several species and different consumption functions or maybe other models different to the chemostat. To conclude, it is not surprising the presence of the death collective rate in both conditions (2.63) since it must be difficult to prove the persistence of both species if this parameter is too large. Therefore, the conditions required to get the persistence of both species are, as already pointed out, absolutely reasonable from the biological point of view. 2.3.4 Numerical simulations and final comments In this section we show several numerical simulations for different values of the parameters involved in the random chemostat model with wall growth (2.22)-(2.24) in order to support the results provided 79 Random disturbances on the input flow through this section. As made before, the blue dashed lines will represent the solution of the correspondingdeterministic chemostat modelwhereastheotheronesrepresentdifferentrealizationsofthesolution of the random system. Moreover, we display four different panels in each figure: there is a big one on the left-handsideshowingthegeneral dynamicsofthe modelandtherearethreesmallerpanelsontherighthand side where the individual dynamics of the substrate and both species will be presented. Firstly, we present several cases in which either the persistence of both species or the extinction is obtained and, eventually, we make a comparison between the random chemostat model with wall growth (2.22)-(2.24) and the one by perturbing the input flow by means of the standard Wiener process, which was already analyzed in Section 1.2 in Chapter 1, in order to show the differences between both ways of modeling the disturbances on the input flow in the chemostat model and remark the important advantages of using a bounded noise, in particular, the Ornstein-Uhlenbeck process. Ontheonehand,weshowsomenumericalsimulationsconcerningtherandomchemostatmodelwith wall growth (2.22)-(2.24). We consider in Figure 2.15 the following values of the different parameters involved in the model: sin =4,D=2,a=1.6,m=2,b=0.5,ν=1.2,c=3,r1=0.2,r2=0.4,α=0.5, β=1,ν=0.2 andwe willtake s0=2.5,x10 =2,x20 =2as initialvaluesforthesubstrateandboth species, respectively. As a result, we can see that both species persist. Figure 2.15: Persistence of the species in the random chemostat model with wall growth 80 Random disturbances on the input flow In the sequel, we only refer to the parameters to be changed respect to the last ones used and we suppose the rest to be the same than before. For instance, in Figure 2.16 we will increase the quantity of noise to α=2, the mean reversion constant to β=4and the volatility constant to ν=0.7, respect to the parameters used in the last figure. We can observe that, even though the intensity of the noise is considerable large, we canstill obtain the persistence of both species and we also remark that it could be inconceivable toconsidersuchalargeintensityofnoisewhenusingthestandardBrownianmotionwithoutobtainingimportant drawbacks which would be, as explained in Chapter 1, totally unrealistic from the biological point of view. Figure 2.16: Persistence of the species in the random chemostat model with wall growth (ii) On another hand, in Figure 2.17 we take sin =4,D=1.5,a=1.6,m=2,b=1,ν=1.7,c=2.4, r1=0.6,r2=0.4,α=0.5,β=1,ν=0.2 and we will take s0=2.5,x10 =2,x20 =2as initial values for the substrate and both species, respectively. Then, both species become extinct, as canbe easily observed, which is quite logical in view of the values of the parameters, specially the collective death rate and the consumption rate of the species, which have been increased respect to the case of persistence. 81 Random disturbances on the input flow Figure 2.17: Extinction of the species in the random chemostat model with wall growth Finally, in Figure 2.18 we increase the quantity of noise to α=2, the mean reversion constant to β=4 and the volatility constant to ν=0.7 and we remark that the rest of the parameters do not change respect to the last ones in Figure 2.17. In this case we can also see easily that both species become extinct, which is notsurprisingbytakingintoaccountwhathappenedinthelastfigureandthenewvaluesoftheparameters involved in the disturbances. Figure 2.18: Extinction of the species in the random chemostat model with wall growth (ii) 82 Random disturbances on the input flow Eventually,wepresentsomenumericalsimulationswhereatypicalrealizationofthestochasticchemostat model with wall growth, which was already analyzed in Section 1.2 in Chapter 1, and two typical ones of the random chemostat model with wall growth will be plotted together in order to see easily the differences of modeling the disturbances on the input flow in the chemostat model by using both the white noise (orange lines) and the Ornstein-Uhlenbeck process (red and green lines). Firstly, in Figure 2.19 we take sin =4,D=2,a=1.6,m=2,b=0.5,ν=1.2,c=3,r1=0.2,r2=0.4 and we consider s0=2.5,x10 =2,x20 =2as initial values for the substrate and both species, respectively. We also choose α=0.8,ν=0.7 and both β=2(red line) and β=1(green line). In this case, we can see that both species persist and we remark the huge disturbances obtained in case of using the white noise respect to the Ornstein-Uhlenbeck process, in fact, these disturbances can be also observed to affect to the specie x2eventhoughitisnotaffectedbytherandominputflowdirectly(see(2.24)),whichdonothappen when perturbing the input flow bymeans of the bounded noise. 2 1.5 x1(t) 1 Phase plane 0.5 3 2 1 s(t) 0 -1 6 5.5 5 4.5 4 3.5 3 2.5 2 x2(t) t (time) 0 5 10 15 20 25 30 35 40 45 50 s(t) -2 -1 0 1 2 3 4Substrate t (time) 0 5 10 15 20 25 30 35 40 45 50 x1(t) 0 0.5 1 1.5 2 2.5 Specie 1 (medium) t (time) 0 5 10 15 20 25 30 35 40 45 50 x2(t) 2 3 4 5 6 7Specie 2 (walls) Figure 2.19: Comparison in case of persistence Finally, inFigure2.20we take sin =4,D=1.5,a=1.6,m=2,b=1,ν=1.7,c=2.4,r1=0.6,r2=0.4 and we consider s0=2.5,x10 =2,x20 =2as initial values for the substrate and both species, respectively. Inthiscaseweincreasethequantityofnoise,respecttothelastcase,toα=1.5,ν=0.7andbothβ=2(red) and β=1(green). We can observe that both species become extinct and we remark again the significant disturbances when using the white noise respect to the case when considering the Ornstein-Uhlenbeck process. 83 Environmental effects by means of the white noise andF:X×Ω−→ R2is given by F(η,ω)=   ma a+η1e−αz∗(ω)η2+Dsineαz∗(ω) −ma a+η1e−αz∗(ω)η2  , where η=(η1,η2)∈X. Since t7→ z∗(θtω)is continuous, Lgenerates an evolution system on R2. Moreover, we notice that ∂ ∂η1·±ma a+η1e−αz∗η2+¸= ∓ mae−αz∗ (a+η1e−αz∗)2η2, and ∂ ∂η2·±ma a+η1e−αz∗η2+¸= ± ma a+η1e−αz∗, wheredenotesaconstantwhichdoesnotdependson(η1,η2)∈X,thereforeF(·,θtω)∈C(X×[0,∞);R2) and is continuously differentiable with respect to the variables (η1,η2), which implies that it is locally Lipschitz with respect to (η1,η2)∈X. Therefore, thanks to classical results from the theory of ordinary differential equations, system (3.9)- (3.10) possesses a unique local solution. Let us checknow that in fact this solution is a global one. We define q(t) :=σ(t)+κ(t)and thanks to (3.7)-(3.8) we have dq dt = − (¯ D+αz∗)q+sinDeαz∗. By solving the previous differential equation we obtain q(t;0,ω,q0)=q0e−¯ Dt−αRt 0z∗ds +sinDZt 0 eαz∗e−¯ D(t−s)−αRt sz∗dr ds,(3.11) hence qis clearly bounded above byan expression which does not blow up. On the other hand, from (3.7) dσ dt ≤ − (¯ D+αz∗)σ+sinDeαz∗. Hence, similarly to previous calculations, σ(t;0,ω,σ0)≤σ0e−¯ Dt−αRt 0z∗ds +sinDZt 0 eαz∗e−¯ D(t−s)−αRt sz∗dr ds, 90 Environmental effects by means of the white noise thus σdoes not blow up either. Summingup,we haveprovedthat q(t)andσ(t)donotblowupandthesamehappenstoκ(t)=q(t)− σ(t). Therefore, the unique local solution to system (3.7)-(3.8) can be extended to a unique global one. Now we would like to check that the global solution of (3.7)-(3.8) belongs to the first quadrant for any t≥0. From (3.7), if σ(t)=0for some t∗∈R+, we have dσ dt (t∗)="−(¯ D+αz∗)σ−mσe−αz∗ a+σe−αz∗κ+sinDeαz∗#(t∗)=sinDeαz∗>0. Besides, given (σ0,0) with σ0>0, there exists a unique solution of system (3.7)-(3.8) satisfying σ(t0)=σ0 andκ(t0)=0for some initial time t0≥0. Imposing κ≡0we deduce that σis given by σ(t;t0,ω,σ(t0)) =σ(t0)e−¯ D(t−t0)−αRt t0z∗ds +sinDZt t0 eαz∗e−¯ D(t−s)−αRt sz∗dr ds. Now, let us pick (σ0,κ0)∈X. Thus, there exists a unique solution (σ(t),κ(t)) such that σ(0) =σ0and κ(0) =κ0. If there is some first t∗>0verifying κ(t∗)=0, thenwe have that (σ(·),κ(·)) is the unique solution of system (3.7)-(3.8) with σ(t∗)=σ∗and κ(t∗)=0. Moreover κ(t)>0for all 0≤t<t∗; however, we already have another solution (σ(t),0) for all t≥t∗−δ(for any δ>0small enough) for this problem, so we obtain a contradiction. As a result, we deduce that for any initial data u0∈Xthe solution remains in the first quadrant. Now we can define the mapping ϕu:R+×Ω×X→Xby ϕu(t,ω)u0:=u(t;0,ω,u0), for all t≥0, u0∈X,ω∈Ω. SincethefunctionFiscontinuousin(u,t), andismeasurableinω,weobtainthatthepreviousmappingis (B[0,∞)×F×B(X),B(X))−measurable. It then follows that (3.7)-(3.8) generate the continuous random dynamical system{ϕu(t,ω)}t≥0,ω∈Ω. 3.1.3 Existence of the random pullback attractor In this section we prove the existence and uniqueness of the random pullback attractor associated to the random chemostat (3.7)-(3.8), describing its internal structure explicitly. Let us remember that E(X)denotes the set of all tempered sets in X. Proposition 3.1.1 For any ε>0, there exists a tempered compact random absorbing set Bε(ω)∈E(X)of the random dynamical system {ϕu(t,ω)}t≥0,ω∈Ω. 91 Environmental effects by means of the white noise Proof. Remember that q=σ+κ. Then, by replacing ωby θ−tωin (3.11), we have q(t;0,θ−tω,q0)=q0e−¯ Dt−αR0 −tz∗(θsω)ds +sinDZt 0 eτh−¯ D+αz∗(θ−τω) τ−α τR0 −τz∗(θrω)dr idτ, andtherefore lim t→+∞ q(t;0,θ−tω,q0)=sinDρ∗(ω) since ¯ Dis always positive, where ρ∗(ω)is definedby ρ∗(ω) :=Z∞ 0 eτh−¯ D+αz∗(θ−τω) τ−α τR0 −τz∗(θrω)dr idτ.(3.12) Note that the above integrand converges to zero when τgoes to infinity, but not the integral. Moreover, ρ∗(ω)has sub-exponential growth. Therefore, for any given ε>0andeach u0∈E(θ−tω), there exists TE(ω,ε)>0such that sinDρ∗(ω)−ε≤q(t;0,θ−tω,q0)≤sinDρ∗(ω)+ε for all t≥TE(ω,ε). We now define Bε(ω) :=©(σ,κ)∈X:sinDρ∗(ω)−ε≤σ+κ≤sinDρ∗(ω)+εª, thus Bε(ω)∈E(ω)is a tempered compact random absorbing set in Xfor any ε>0. Hence, from Proposition B.0.1 in Appendix B, the random dynamical system generated by the system (3.7)-(3.8) possesses a unique random attractor given by A={A(ω)}ω∈Ω⊂Bε(ω)for any ε>0. Thus A= {A(ω)}ω∈Ω⊂B0(ω), i.e., we have the following expression for each component of the attractor A(ω) :=(sinDρ∗(ω)−κ(ω),κ(ω)). Thefollowingresultprovidesinformationconcerningtheinternalstructureoftheuniquerandompullback attractor associated to the random chemostat (3.7)-(3.8). Proposition 3.1.2 For ¯ Ddefined by (3.5) assume that ¯ D>m.(3.13) Then, the random attractor associated to the random dynamical system {ϕu(t,ω)}t≥0,ω∈Ωhas the following structure: A={A(ω)}ω∈Ω,where A(ω)=(sinDρ∗(ω),0). 92 Environmental effects by means of the white noise Proof. Thanks to (3.8) we knowthat dκ dt ≤ −(¯ D−m+αz∗)κ, whose solution, after replacing ωby θ−tωand making tgo to infinity, tends to zero provided ¯ D>m, thus theinternalstructureoftheattractorinthiscaseconsistsofsingletonsubsets A(ω)=(sinDρ∗(ω),0)which means that there is not persistence of the microorganism. We would like to remark that it is not possible to ensure the persistence of the microorganism in case ¯ D≤mby using mathematical arguments even though our simulations show that the random attractor in this case is totally contained in X, in other words, our model seems to guarantee the persistence of the microorganism. 3.1.4 Random pullback attractor for the stochastic system We have proved that the system (3.7)-(3.8) has a unique globalsolution u(t;0,ω,u0)which remains in X, the first quadrant, forany u0∈Xand generates a random dynamical system {ϕu(t,ω)}t≥0,ω∈Ω. Now, we define a mapping T:Ω×X−→ X as follows T(ω,ζ)=³ζ1eαz∗(ω),ζ2eαz∗(ω)´, whose inverse is given by T−1(ω,ζ)=³ζ1e−αz∗(ω),ζ2e−αz∗(ω)´. We know that v(t)=(s(t),x(t)) and u(t)=(σ(t),κ(t)) are related by (3.6). Since Tis a homeomorphism, thanks to Lemma B.0.1 in Appendix B we obtain a conjugatedrandomdynamical system given by ϕv(t,ω)v0:=T−1(θtω,ϕu(t,ω)T(ω,v0)) =T−1(θtω,ϕu(t,ω)u0) =T−1(θtω,u(t;ω,u0)) =v(t;ω,v0), whichmeansthat {ϕv(t,ω)}t≥0,ω∈Ωisa randomdynamicalsystemforouroriginal stochasticsystem(3.3)- (3.4). Moreover, thanks to Lemma B.0.2 in Appendix B, the random pullback attractor of the random system without taking into account the wall growth (3.7)-(3.8), A={A(ω)}ω∈Ω⊂B0(ω), becomes into AT= 93 Environmental effects by means of the white noise {AT(ω)}ω∈Ω⊂BT 0(ω), the random pullback attractor of the system (3.3)-(3.4), where BT 0(ω) :=n(s,x)∈X:s+x=Dsinρ∗(ω)e−αz∗(ω)o. In other words, each component of our random pullback attractor, AT(ω), canbe writtenas AT(ω) :=³sinDρ∗(ω)−se−αz∗(ω),se−αz∗(ω)´. In addition, we know that the internal structure of the attractor consists of singleton subsets AT(ω)= ¡sinDρ∗(ω)e−αz∗(ω),0¢as long as ¯ D>m. Apart from that, it is not possible to ensure the persistence of the microorganism otherwise even though our simulations show the persistence for several values of the parameters (see Section 3.1.5). 3.1.5 Numerical simulations and final comments In this section we will show some numerical simulations concerning the results provided through the firstpartofthischapterbyusingtheEuler-Maruyama'smethodalreadydescribedinChapter1. Inaddition, ineachfigurethebluedashedlinesrepresentthesolutionofthedeterministicchemostatmodel(3.1)-(3.2) whereas the rest represent different realizations ofthe stochastic chemostat (3.3)-(3.4). Henceforth, we consider sin =1,a=0.6,m=3and we take s0=2.5 and x0=5as initial pair. In addition, we specify the value of both the dilution rate and also the intensity of the noise which will help us to obtaindifferentsituationsshowingeithertheextinction orthepersistence ofthespecies,dependingon the conditions previously given from the mathematical study. On the one hand, in Figure 3.1 we consider D=3and we take α=0.1 (left) and α=0.5 (right). It is easy to notice that the species become extinct in both cases, which is not surprising since condition (3.13) in Proposition 3.1.2is fulfilled. Apart fromthat, the difference between bothpicturescanbe noticeddue to the effects caused by a larger quantity of noise in the second one. 94 Environmental effects by means of the white noise s(t) 0 0.5 1 1.5 2 2.5 x(t) 0 1 2 3 4 5 6Phase plane s(t) 0 0.5 1 1.5 2 2.5 3 x(t) 0 1 2 3 4 5 6 7 8Phase plane Figure 3.1: Extinction. α=0.1 (left) and α=0.5 (right) Now, in Figure 3.2 we increase the intensity ofthe noise to α=1(left) and α=1.5 (right). In this case, sincecondition(3.13)inProposition3.1.2isalsotrue,thespeciesbecomeextinctinbothcasesandtheeffect of the different values of the quantify of noise can be also easily noticed. s(t) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x(t) 0 1 2 3 4 5 6 7Phase plane s(t) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x(t) 0 2 4 6 8 10 12 Phase plane Figure 3.2: Extinction. α=1(left) and α=1.5 (right) On the other hand, in Figure 3.3 we consider D=1and we take α=0.1 (left) and α=0.5 (right). In this case, condition (3.13) in Proposition 3.1.2 does not hold true, in fact, the persistence of the microbial biomass is obtained for the previous values of the parameters. 95 Environmental effects by means of the white noise s(t) 0 0.5 1 1.5 2 2.5 x(t) 0 1 2 3 4 5 6 7Phase plane s(t) 0 0.5 1 1.5 2 2.5 3 x(t) 0 1 2 3 4 5 6 7 8 9Phase plane Figure 3.3: Persistence. α=0.1 (left) and α=0.5 (right) Finally, in Figure 3.4 we consider D=1again and we increase the intensity of the noise to α=1(left) and α=1.5 (right). In this case condition (3.13) in Proposition 3.1.2 is not fulfilled but, as explained when making the mathematical analysis, it is not possible to guarantee the persistence of the microorganisms, specially if the quantity of noise is considerable large. s(t) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x(t) 0 2 4 6 8 10 12 14 16 18 20 Phase plane s(t) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x(t) 0 1 2 3 4 5 6 7 8 9 10 Phase plane Figure 3.4: Extinction. α=1(left) and α=1.5 (right) We would like to emphasize the important differences when comparing the previous numerical simulations shown in this section with the ones presented in Chapter 1. In this case every realization of the solution of the stochastic chemostat model remains positive, as proved from the mathematical point of view in Theorem 3.1.1, independently of the initial values and, what is more, the intensity of the noise. It is mainly due to the way of modeling the stochasticity now and it is also a clear proof that this way is much more realistic from the biological point of view than the one used in Chapter 1. 96 Environmental effects by means of the white noise 3.2 Stochastic chemostat model with wall growth In this section we analyze the equivalent model with wall growth. Then, let us now introduce the simplest chemostat model with wall growth ds dt =D(sin −s)−ms a+sx1−ms a+sx2+bνx1,(3.14) dx1 dt = −(ν+D)x1+cs a+sx1−r1x1+r2x2,(3.15) dx2 dt = −νx2+cs a+sx2+r1x1−r2x2,(3.16) where s(t),x1(t)and x2(t)denote concentrations of the nutrient and the two different microorganisms, respectively;b∈(0,1) describesthefractionofdeadbiomasswhichisrecycled,ν>0isthecollectivedeath ratecoefficient,r1>0andr2>0represent the rates at whichthe speciesstickon toandshear offfromthe walls, respectively, and 0<c≤mis the growth rate coefficient of the consumer species. By introducing again a white noise in each equation of (3.14)-(3.16) and using the conversion between It¯ o and Stratonovich interpretations, we finally obtain the following stochastic system with wall growth ds =h−¯ Ds +bνx1−ms a+sx1−ms a+sx2+Dsinidt −αs◦dW (t), (3.17) dx1=h−¡ν+¯ D+r1¢x1+cs a+sx1+r2x2idt −αx1◦dW (t), (3.18) dx2=·r1x1−µν+r2+α2 2¶x2+cs a+sx2¸dt −αx2◦dW (t). (3.19) 3.2.1 Stochastic chemostat becomes a random chemostat In this section we will analyze the stochastic chemostat model with wall growth (3.17)-(3.19) by performing a variable change which involves, as in the previous section, the well-known stationary Ornstein- Uhlenbeckprocess z∗. To this end, we will firstly define the new variables σ,κ1andκ2as follows σ(t)=s(t)eαz∗(θtω),κ1(t)=x1(t)eαz∗(θtω)and κ2(t)=x2(t)eαz∗(θtω).(3.20) We will write z∗insteadof z∗(θtω)and σ,κ1andκ2in place of σ(t),κ1(t)and κ2(t). From (3.17)-(3.19), bydifferentiation, we obtain the following random system with wall growth dσ dt = − (¯ D+αz∗)σ+bνκ1−mσe−αz∗ a+σe−αz∗(κ1+κ2)+Dsineαz∗,(3.21) 97 Environmental effects by means of the white noise dκ1 dt = − (ν+¯ D+r1+αz∗)κ1+cσe−αz∗ a+σe−αz∗κ1+r2κ2,(3.22) dκ2 dt =r1κ1−µν+r2+α2 2+αz∗¶κ2+cσe−αz∗ a+σe−αz∗κ2.(3.23) 3.2.2 Random chemostat generates a random dynamical system In this section we prove that the random chemostat with wall growth (3.21)-(3.23) possesses a unique global solution which, moreover, generates a random dynamical system. In the sequel, we will denote X:=©(x,y,z)∈R3:x≥0, y≥0, z≥0ª, the first octant. Theorem 3.2.1 For any ω∈Ωand any initial value u0:=(σ0,κ10,κ20)∈X, where σ0,κ0,κ10 and κ20 denote σ(0;0,ω,u0),κ(0;0,ω,u0),κ1(0;0,ω,u0)andκ2(0;0,ω,u0),respectively,therandomsystem(3.21)-(3.23)possesses a unique global solution u(·;0,ω,u0) :=(σ(·;0,ω,u0),κ1(·;0,ω,u0),κ2(·;0,ω,u0)) ∈C1([0,+∞),X) withu(0;0,ω,u0)=u0. Inaddition, thesolution mappinggeneratesa randomdynamical ϕu:R+×Ω×X→X defined as ϕu(t,ω)u0=u(t;0,ω,u0), for all t∈R+,u0∈X,ω∈Ω, the value at time tof the solution of system (3.21)-(3.23) with initial state u0at time zero. Proof. Similarly to the proof of Theorem 3.1.1 and denoting u(·;0,ω,u0) :=(σ(·;0,ω,u0),κ1(·;0,ω,u0),κ2(·;0,ω,u0)), system (3.21)-(3.23) can be written as du dt =L(θtω)u+F(u,θtω), where L(θtω)=    −(¯ D+αz∗)bν−m−m 0−(ν+¯ D+r1+αz∗)+c r2 0r1−³ν+r2+α2 2+αz∗´+c     andF:X×Ω−→ R3is given by F(η,θtω)=       Dsineαz∗+ma a+η1e−αz∗(η2+η3) −ca a+η1e−αz∗η2 −ca a+η1e−αz∗η3        , 98 Environmental effects by means of the white noise where η=(η1,η2,η3)∈X. Sincet7→ z∗(θtω)iscontinuous,LgeneratesanevolutionsystemonR3. Inaddition,F(·,θtω)∈C(X× [0,+∞);R3)andiscontinuouslydifferentiablewithrespecttothevariables(η1,η2,η3),whenceitislocally Lipschitz with respect to (η1,η2,η3)∈X. Therefore, thanks to classical results from the theory of ordinary differential equations, system (3.21)- (3.23) possessesa uniquelocal solution. Letus checknow that, infact, this solution isa global one. In order to do that, we define q(t) :=σ(t)+κ1(t)+κ2(t), which satisfies the following differential equation dq dt ≤ −µα2 2+αz∗¶q+Dsineαz∗,(3.24) since c≤mandb<1. By solving (3.24), we obtain q(t;0,ω,q0)≤q0e−α2 2t−αRt 0z∗ds +Dsin Zt 0 eαz∗eα2 2s+αRs 0z∗dr e−α2 2t−αRt 0z∗dr ds,(3.25) thus qis clearly bounded by above byan expression which does not blow up at any finite time. Furthermore, we have the following differential inequatily dq dt ≥ −(¯ D+αz∗+m+ν)q+Dsineαz∗, therefore we obtain the following bound, which will be further very useful q(t;0,ω,q0)≥q0e−(¯ D+m+ν)t−αRt 0z∗ds +Dsin Zt 0 eαz∗e(¯ D+m+ν)s+αRs 0z∗dr e−(¯ D+m+ν)t−αRt 0z∗dr ds.(3.26) It is straightforward to verify, similarly to the case without taking into account the wall growth, that the global solution of the random system (3.21)-(3.23) belongs to the first octant for any initial data u0∈X and t≥0. To this end, we will consider κ1≥0and κ2≥0and we will evaluate the equation (3.21) when σ=0such that we obtain dσ dt ¯¯¯¯σ=0 =bνκ1+Dsineαz∗>0. Moreover, bytaking σ≥0and κ2≥0and evaluating the equation (3.22) when κ1=0, we have dκ1 dt ¯¯¯¯κ1=0 =r2κ2>0. 99 Environmental effects by means of the white noise 3.2.4 Random pullback attractor for the stochastic chemostat In this section, we define a random dynamical system associated to the original stochastic chemostat model with wall growth by using the conjugation Lemma B.0.1 in Appendix B. In addition, we will provide information about the pullback random attractor associated to the original stochastic system as well as its internal structure. Hence, we define a mapping T:Ω×X−→ X, similarly to the one in Section 3.1, as follows T(ω,ζ)=³ζ1eαz∗(ω),ζ2eαz∗(ω),ζ3eαz∗(ω)´, whose inverse is given by T−1(ω,ζ)=³ζ1e−αz∗(ω),ζ2e−αz∗(ω),ζ3e−αz∗(ω)´. Since v(t)=(s(t),x1(t),x2(t)) and u(t)=(σ(t),κ1(t),κ2(t)) are both related by (3.20) and Tis a homeomorphism, thanks to Lemma B.0.1 in Appendix B we obtain again a conjugated random dynamical system given by ϕv(t,ω)v0:=T−1(θtω,ϕu(t,ω)T(ω,v0)) =T−1(θtω,ϕu(t,ω)u0) =T−1(θtω,u(t;ω,u0)) =v(t;ω,v0), whichmeansthat{ϕv(t,ω)}t≥0,ω∈Ωisarandomdynamicalsystemforouroriginalstochasticsystem(3.17)- (3.19). Inaddition,thankstoLemmaB.0.2inAppendixB,therandompullbackattractoroftherandomsystem with wall growth (3.21)-(3.23), c A={b A(ω)}ω∈Ω⊂b B0(ω), becomes now into c AT={b AT(ω)}ω∈Ω⊂b BT 0(ω), the random pullback attractor associated to the original stochastic system with wall growth (3.17)-(3.19), where b BT 0(ω)=n(s,x1,x2)∈X:Dsinρ∗ l(ω)e−αz∗(ω)≤s+x1+x2≤Dsinρ∗ u(ω)e−αz∗(ω)o. Table 3.2 in the next page shows information on the random pullback attractor c AT={b AT(ω)}ω∈Ω, taking into account the analysis carried out at the end of Section 3.2.3. 106 Environmental effects by means of the white noise ASYMPTOTIC BOUNDS ATTRACTORINTERNAL STRUCTURE Case A: bνcξ−m≥0 (A-1) ν+α2 2>c lim t→+∞ S(t)≥sinDρ∗ σ(ω)e−αz∗(ω)−ε sinDρ∗ σ(ω)e−αz∗(ω) S x1 x2 sinDρ∗ l(ω)e−αz∗(ω) sinDρ∗ u(ω)e−αz∗(ω) lim t→+∞ [x1(t)+x2(t)]≤ε (A-2) ν+α2 2<c lim t→+∞ S(t)≥sinDρ∗ σ(ω)e−αz∗(ω)−ε sinDρ∗ l(ω)e−αz∗(ω) sinDρ∗ u(ω)e−αz∗(ω) S x1 x2 sinDρ∗ σ(ω)e−αz∗(ω) x1+x2does not provide any extra information Case B: bνcξ−m<0 (B-1) ν+α2 2>c lim t→+∞ S(t)≥sinDρ∗ σ(ω)e−αz∗(ω)−ε sinDρ∗ σ(ω)e−αz∗(ω) S x1 x2 sinDρ∗ l(ω)e−αz∗(ω) sinDρ∗ u(ω)e−αz∗(ω) lim t→+∞ [x1(t)+x2(t)]≤ε (B-2) ν+α2 2<c S x1 x2 sinDρ∗ l(ω)e−αz∗(ω) sinDρ∗ u(ω)e−αz∗(ω) Sdoes not provide any extra information x1+x2does not provide any extra information Table 3.2: Internal structure of the random attractor - Stochastic chemostat model with wall growth 107 Environmental effects by means of the white noise 3.2.5 Numerical simulations and final comments In this section we show some numerical simulations concerning the stochastic chemostat model with wall growth (3.17)-(3.19) analyzed in Section 3.2. In each figure we display four different panels: there is a bigoneontheleftshowingthephaseplanewiththegeneraldynamicsofthenutrientandthemicroorganisms;ontheotherhand,threesmallerpanelscanbeseenontherighttoshowthedynamicsofthesubstrate andthedifferentspeciesindividually,dependingonthetime. Inthisway,wecaneasilyobservetheasymptotic behavior of every state variable involved in our chemostat. We also recall that the blue dashed lines represent the solution of the deterministic chemostat with wall growth (3.14)-(3.16) whereas the rest represent different realizations of the stochastic one given by (3.17)-(3.19). Moreover, the thick black asterisk denotes the initial value (s0,x10,x20). On the one hand, in Figure 3.5 we consider sin =1,D=2,a=0.6,m=5,b=0.5,ν=2,c=1.2, r1=0.2,r2=0.8 and we take s0=5,x10 =10 and x20 =10 as initial triple. In addition, we consider α=0.1 as intensity of the noise. As a result, we canobserve that both species, the ones in the medium and also the ones sticked on the walls of the culture vessel, become extinct, which is not surprising due to the fact that condition (3.36) holds true. Figure 3.5: Extinction. α=0.1 InFigure3.6weincreasethequantityofthenoisetoα=0.5. Inthiscasecondition(3.36)isalsofulfilled, thus we obtain the extinction of the species. 108 Environmental effects by means of the white noise Figure 3.6: Extinction. α=0.5 On the other hand, in Figure 3.7 we consider sin =1,D=2,a=0.6,m=5,b=0.5,ν=0.3,c=3, r1=0.2,r2=0.8, we take s0=5,x10 =10 and x20 =10 as initial triple and we consider α=0.1. In this casecondition(3.36)doesnotholdtrue,infact,themicrobialbiomasspersistseventhoughitisnotpossible to ensure the persistence from the mathematical point of view. Figure 3.7: Persistence. α=0.1 Finally, in Figure 3.8 we increase the quantity of the noise to α=0.5. In this case, condition (3.36) is notfulfilledeitherbut,althoughthepersistencecannotbe guaranteedaswepreviouslydeducedfromthe 109 Environmental effects by means of the white noise mathematical study, both species, the ones in the medium and also the ones sticked on the walls of the culture vessel, persists, as can be easily observed in the last two panels. Apart from that, in this figure we canalsonoticethe increment ofthe quantityof the noise whenlookingatthe differentrealizations,which are more disturbed that the ones in the previous figure with α=0.1. Figure 3.8: Persistence. α=0.5 We would like to highlightthat every realization in each figure of this section remains positive for any initial value and every value of the parameters involved in the system, as proved in Theorem 3.2.1. Inaddition, it is not strange the fact of having extinction in Figures 3.5 and 3.6 whereas we obtain the persistence in Figures 3.7 and 3.8. The most important reason, apart from the intensity of the noise, is the value of the collectivedeathcoefficientwhichismuchsmallerinthelasttwofiguresanditdirectlyhasahugeinfluence on the condition (3.36), thanks to which the extinction can be (or not) ensured, independently of the other parameters. 110 C h a p t e r 4 Modeling and analysis of environmental effects caused by the fractional Brownian motion in chemostat models In this last chapter we analyze the chemostat model after introducing a stochastic perturbation given by means of a new kind of noise: the fractional Brownian motion BHindexed by the Hurst parameter H. Firstly, wewillintroducethisnewnoiseandwewillmentionitsmainproperties. Sincethefractional Brownian motionfor H6= 1/2 is not a semimartingale, we cannot use the It¯ o integral in this setting, hence we need to introduce a new integral against BH. To be more precise, we define the stochastic integral by using fractional calculus tools, for what the main ingredient is the use of fractional derivatives. The main advantage of using this definition is that it is a pathwise integral and, therefore, it does not produce exceptional sets which are against the generation of a cocycle. Then, we will establish a result concerning the existence and uniqueness of a global solution of our stochastic system and will provide some conditions under which our system generates a semigroup operator, which allows us to define the solution in a mild sense. Since we do not transform the stochastic chemostat model perturbed by BHinto a random system (as we already did in the previous chapters when dealing with B1 2), this means that the norm of the solution depends in particular of the norm of the noise, which prevents us to use a suitable version of Gronwall's lemma to conclude, for instance, the existence of an absorbing ball. This is the main reason for whichwewillintroduceasequenceofstoppingtimes,whichwillallowustocontrolthecontributionofthe noise,and,therefore,toensureinparticulartheexistenceofa(discrete)temperedabsorbingsetwhichwill be determinant to establish the existence and uniqueness of a (discrete) random pullback attractor. We would like to emphasize that, in a first step, we discretize the random dynamical system, in a way that we onlyconsideritactingonthesequenceofstoppingtimes. Indeed,whenwelookatthecocyclemappingon the stopping times we can handle the estimates of the solution. In a second step, we will obtain the exis- 111 Environmental effects caused by the fractional Brownian motion tence and uniqueness of the continuous random pullback attractor associated to the continuous random dynamical system, which rests upon the relationship between the discrete and continuous tempered sets and the properties of the stopping times. Finally, we will show some numerical simulations to observe the effects of the fractional Brownian motion in the chemostat models previouslystudied. The results and explanations concerning this chapter can be found in [11]. 4.1 Chemostat models and fractional noise In this section we will consider the simplest deterministic chemostat model (3)-(4), which was already presentedinthe introductory chapter,andwe willintroducestochasticityby using a differentkindof noise which significantly differs from the standard Brownian motion used in Chapters 1 and 3. This new noise, that is becoming progressively more and more popular between researchers of several branches of scientific knowledge, is the fractional Brownian motion, fBm for short, that was introduced originally by Kolmogorov in 1940 (see [57]) to study the long term storage capacity of reservoirs along the Nile river. The fBm consists of a generalization of the standard Brownian motion, as we will see in Section 4.2. The main reasonto introduce this new noise in our model is due to the fact that it has been proved to be a good candidate to model random long-time influences in climate systems, hydrology and medicine, to name a few applications. Similarly to the way of modeling in Chapter 3, when using the standard Wiener process, we will introduce some disturbances in the deterministic model (3)-(4) by using the fBm such that we obtain the following stochastic differential system ds =h(sin −s)D−msx a+sidt +αsdBH(t), (4.1) dx =h−Dx +msx a+sidt +αxdBH(t), (4.2) where BH(·)denotes the fBm with Hurst parameter H∈(1/2,1) and α>0refers to the quantity of noise, s(t)and x(t)denote concentrations of the nutrient and the microbial biomass, respectively; sin denotes the volumetric dilution rate, ais the half-saturationconstant, Dis the dilution rate and mis the maximal consumption rate of the nutrient and also the maximal specific growth rate of microorganisms. We notice thatallparametersarepositiveandweuseafunctionHollingtype-IIasfunctionalresponseofthemicroor- ganism describing howthe nutrient is consumed by the species. Up to now, wehavejust mentionedthe chemostat modelwithout taking into accountthe wallgrowth. Ofcoursethatwecanalsoconsidertheequivalentstochasticmodelwithwallgrowth,whichwasalreadyintroducedintheintroductorychapter,affectedbyanfBmaswell,givenbythefollowingsystemofstochastic 112 Environmental effects caused by the fractional Brownian motion differential equations ds =h(sin −s)D−ms a+sx1−ms a+sx2+bνx1idt +αsdBH(t), (4.3) dx1=h−(ν+D)x1+cs a+sx1−r1x1+r2x2idt +αx1dBH(t), (4.4) dx2=h−νx2+cs a+sx2+r1x1−r2x2idt +αx2dBH(t), (4.5) whereBH(·)denotesagainthefBmwithHurstparameterH∈(1/2,1)andtherestofparametersandfunctions have the same meaning than in system (6)-(8). From now on, we will focus our study just on the simplest stochastic chemostat model (4.1)-(4.2) since the same analysis could be made for the model with wall growth (4.3)-(4.5). 4.2 The fractional Brownian motion and the fractional integral The fractional Brownian motion is a centered Gaussian process BH={BH(t)}t∈Rindexed bythe Hurst parameter H∈(0,1). In fact, it satisfies the long-memory property when H>1/2. In addition, it is a selfsimilar stochastic process characterized by the stationarity of its increments. When H=1/2, the fBm reducestothestandardBrownianmotion. Moreover,ithascontinuoussamplepathsanditscovariancefunction is given by E¡BH(s),BH(t)¢=1 2¡t2H+s2H− |t−s|2H¢. ThankstoKolmogorov'sTheorem,thereexistsacontinuousversionofthefractionalBrownianmotion, whose canonical interpretation is as follows. Let Ω=C0(R,R)be the space of continuous functions on R with values in Rwhich are zero at zero. This set is equipped with the compact open topology. We define Fas the Borel σ−algebra and Pis the probability measure, or probability distribution, of the fBm. Then, thequadruple(Ω,F,P,{θt}t∈R)is anergodicmetricdynamical system,where{θt}t∈Rdenotesthe Wiener shiftdefinedasinAppendixA.Formoredetailedinformation,see[39]. Thiscanonicalprocesshasaversion, which will be denoted byω, which is β−Hölder continuous on any interval [−k,k]⊂Rfor β<H, see [58]. WhenH6= 1/2,thefBmbehavesinaverydifferentwaythanthestandardBrownianmotion. Infact,BH is neither a martingale nor a Markov process hence we cannot use the It¯ o's theory to define the stochastic integralwithintegratorthefBm. Thishasthedisadvantagethatweneedtodefineadifferentintegralwith respect to BH, but, as a positive counterpart, the so-called pathwise integral does not produce exceptional sets. Letusremindthat,asfarasthecocyclepropertyisconcerned,exceptionalsetsarenotpermitted. Asit iswell-known,theIt¯ o'sintegralproducesexceptionalsetsandthisisthereasonforwhichinalltheprevious chapters we always transformed the stochastic systems into random ones that were handled by deterministic techniques. In this chapter we will not use anytransformation but instead we deal with the stochastic chemostat with fBm. 113 Environmental effects caused by the fractional Brownian motion Intherestofthesection,wewillintroduce somebasicconceptsconcerningthe theoryofthefractional calculus whichwill involve twoessential ingredients: the fractional derivatives and the fractional integral. From now on, T>0will denote a positive real time. Then, we introduce the following usual Hölder spaces. Definition 4.2.1 Let β>0be a constant and d∈N. Then, Cβ([a,b];Rd)denotes the space of Hölder continuous functions f: [a,b]−→ Rdwith values in Rdequipped with the norm kfkβ,a,b:= kfk∞,a,b+ |||f|||β,a,b =sup t∈[a,b] |f(t)|+ sup a≤s<t≤b |f(t)−f(s)| (t−s)β< ∞. Moreover, we present now the damped Hölder spaces which are given as in the following definition. Definition 4.2.2 Let β>0be a constant and d∈N. Then, Cβ β([a,b];Rd)denotes the space of continuous functions f: [a,b]−→ Rdwith values in Rdequipped with the norm kfkβ,β,a,b:= kfk∞,a,b+ |||f|||β,β,a,b =sup t∈[a,b] |f(t)|+ sup a≤s<t≤b (s−a)β|f(t)−f(s)| (t−s)β< ∞. We would like to notice that the damped Hölder spaces defined in the previous definition are Banach spaces (see [21,62]). Now, we introduce the concept of fractional derivatives which will allow us to define the fractional integral. We refer every interested reader to [78] for more detailed information on fractional calculus. Definition 4.2.3 Let α∈(0,1) be a constant and a<breal numbers. Then, the fractional derivatives of Kand ω, respectively, are defined as follows Dα a+K(r) :=1 Γ(1−α)µK(r) (r−a)α+αZr a K(r)−K(s) (r−s)α+1ds¶, D1−α b−ωb−(r) :=1 Γ(α)µω(r)−ω(b) (b−r)1−α+(1−α)Zb r ω(r)−ω(s) (s−r)2−αds¶, where Γdenotes the Gamma function. It is straightforward to prove the following bound concerning the fractional derivative of the fractional Brownian motion. 114 Environmental effects caused by the fractional Brownian motion Lemma 4.2.1 For every β∈(1/2,H)and a,b,x∈Rsuch that [a,b]⊆[0,T]and x∈[a,b], for α>1−βthe following inequality holds true |D1−α b−ωb−(x)| ≤ c(β)|||ω|||β,0,T(b−x)β+α−1. Now, we introduce the fractional integral. Definition 4.2.4 Let K∈Cβ β([a,b];Rd)such that β>1/2 and 1−β<α<β. Then, we define the fractional integral of Kwith respect to the fBm as follows Zb a K(r)dω(r) :=(−1)αZb a Dα a+K(r)D1−α b−ωb−(r)dr,(4.6) where the fractional derivatives inside the integral on the right hand side of (4.6)are defined as in Definition 4.2.3. For a detailed construction of the pathwise stochastic integral (4.6) as well as its main properties, we refer the reader to [78]. In the following result, nevertheless, we collect some of these properties. Lemma 4.2.2 Given T>0,K∈Cβ β([0,T];Rd),ω∈Ωan fBm with 1−β<α<β. Then, 1) For [a,b]⊆[0,T], we have the following estimate of the fractional integral given by (4.6) ¯¯¯¯Zb a K(r)dω(r)¯¯¯¯≤c(α,β)kKkβ,β,0,T|||ω|||β,0,T(b−a)β.(4.7) 2) For [a,b]⊆[0,T]and p∈Rsuch that [a−p,b−p]⊆[0,T], the following equality holds Zb a K(r)dω(r)=Zb−p a−p K(r+p)dθpω(r). 4.3 Chemostat model driven by the fractional Brownian motion As explained before, we are interested in investigating the stochastic chemostat model (4.1)-(4.2) perturbed by the fractional Brownian motion. To this end, we will firstly linearize our stochastic differential system around (sin,0), the so-called washout equilibrium of the deterministic chemostat model given by (3)-(4), which allows us to rewrite the chemostat model (4.1)-(4.2) in the following abstract form du =(Au +F(u))dt +G(u)dBH(t), (4.8) where u=(s,x), with A= −D−m+ma a+sin 0−D+m−ma a+sin  ,F(u)= Dsin +ma a+Sx−ma a+sin x −ma a+Sx+ma a+sin x ,G(u)= αS αx  115 Environmental effects caused by the fractional Brownian motion +k1 n X m=1 e−λ1(Tn(θTjω)−Tm(θTjω)) +k1.(4.17) Thefollowingnaturalstepconsistsinintroducingadiscreterandomdynamicalsystemwhichisdefined as the restriction of the original continuous random dynamical system to the sequence of stopping times {Ti(ω)}i∈Zgiven by (4.14). More precisely, we firstly introduce the new discrete shift, which is given by the mapping e θ:Z×Z−→ Z (i,j)7−→ e θ(i,j)≡e θij:=i+j that satisfies the following properties: on the one hand, the mapping e θ0is trivially the identity operator in Z; on another hand, the cocycle property e θi1+i2j=(e θi1◦e θi2)j also holds. Now, we introduce the following mapping Φ:Z+×Z×Ω×R2 +−→ R2 +, definedby Φ(i,j,ω,u0) :=ϕ(Ti(θTj(ω)ω),θTj(ω)ω,u0), (4.18) the solution given by (4.10) at time Ti(θTj(ω)ω)for the noise path θTj(ω)ω. We would like to emphasize that ωonly acts as a parameter in (4.18). Apart from that, for the sake of simplicity, we write θTjωinstead of θTj(ω)ωwhen no confusion is possible. It is not difficult to prove that the mapping given by (4.18) is actually a discrete dynamical system. The proof of this statement follows trivially from its definition since the property is inherited from the original continuous random dynamical system {ϕ(t,ω)}t≥0,ω∈Ω. Now, wemake useofthe estimate (4.17) to prove the existence of a discrete absorbing setwhich allows us to conclude that there exists a unique discrete random pullback attractor associate to the discrete dynamical systemdefined by (4.18). Henceforth, we will denote the discrete tempered sets by E(i,ω), where i∈Zand ω∈Ωacts as a parameter. In addition, ρ(i,ω)will denote the radius of the corresponding discrete tempered set previously mentioned. Theorem 4.3.6 Assume that for ω∈Ω, there exists D>0such that λ1D>k1eλ1µ(4.19) 122 Environmental effects caused by the fractional Brownian motion and liminf q→−∞ |Tq(θTjω)| |q|>D(4.20) hold true for the noise path θTj(ω), where j∈Z. Then, there exists a tempered discrete absorbing set associated to the discrete dynamical system (4.18), which is given by the ball centered at the origin with radius R(j,ω) :=2k1 0 X q=−∞ eλ1Tq(θTjω)e−k1qeλ1µ,(4.21) where Tq(·)denotes the q−stopping time for q∈Z. Proof. After multiplying (4.17) by the factor eλ1Tn(θTjω), we have eλ1Tn(θTjω)kukβ,β,Tn(θTjω),Tn+1(θTjω) ≤k0|u0| + k1 n X m=1 eλ1Tm(θTjω)e−λ1Tm−1(θTjω)eλ1Tm−1(θTjω)kukβ,β,Tm−1(θTjω),Tm(θTjω) +k1 n X m=1 eλ1Tm(θTjω)+k1eλ1Tn(θTjω), =k0|u0| + k1 n−1 X m=0 eλ1T1(θTm+jω)eλ1Tm(θTjω)kukβ,β,Tm(θTjω),Tm+1(θTjω) +k1 n−1 X m=0 eλ1Tm+1(θTjω)+k1eλ1Tn(θTjω), whence, considering ykdefinedas yk:=eλ1Tk(θTjω)kukβ,β,Tk(θTjω),Tk+1(θTjω), we have yn≤k0|u0| + k1 n−1 X m=0 eλ1T1(θTm+jω)ym+2k1 n−1 X m=0 eλ1Tm+1(θTjω). Now, we define the positive functions fn:=k0|u0| + 2k1 n X m=1 eλ1Tm(θTjω)and gk:=k1eλ1T1(θTk+jω),(4.22) andconsider ψ0:=k0|u0|and ψq:=2k1eλ1Tq(θTjω).(4.23) Hence, thanks to (4.22) and (4.23), applying the discrete Gronwall lemma C.0.4 (see Appendix C), we 123 Environmental effects caused by the fractional Brownian motion obtain that yn≤k0|u0| n−1 Y l=0³1+k1eλ1T1(θTl+jω)´+2k1 n X q=1 eλ1Tq(θTjω)n−1 Y l=q³1+k1eλ1T1(θTl+jω)´ ≤k0|u0| n−1 Y l=0 ek1eλ1T1(θTl+jω) +2k1 n X q=1 eλ1Tq(θTjω)n−1 Y l=q ek1eλ1T1(θTl+jω) =k0|u0|ek1Pn−1 l=0eλ1T1(θTl+jω) +2k1 n X q=1 eλ1Tq(θTjω)ek1Pn−1 l=qeλ1T1(θTl+jω) ≤k0|u0|ek1neλ1µ+2k1 n X q=1 eλ1Tq(θTjω)ek1(n−q)eλ1µ, where the last inequality holds since T1(θTl+jω)≤µ, then n−1 X l=0 eλ1T1(θTl+jω)≤ n−1 X l=0 eλ1µ=neλ1µand n−1 X l=q eλ1T1(θTl+jω)≤(n−q)eλ1µ. Hence, we obtain the final bound for the norm of our solution, which is given by kukβ,β,Tn(θTjω),Tn+1(θTjω)≤k0|u0|ek1neλ1µe−λ1Tn(θTjω) +2k1 n X q=1 eλ1Tq(θTjω)e−λ1Tn(θTjω)ek1(n−q)eλ1µ. Now, thanks to the definition of the discrete dynamical system (4.18), we have that |Φ(n,j,ω,u0)| ≤ kukβ,β,Tn(θTjω),Tn+1(θTjω) ≤k0|u0|ek1neλ1µe−λ1Tn(θTjω)+2k1 n X q=1 eλ1Tq(θTjω)e−λ1Tn(θTjω)ek1(n−q)eλ1µ.(4.24) By replacing jby e θ−nj=j−nin (4.24) and taking supremum, we obtain sup u0∈E(j−n,ω) |Φ(n,j−n,ω,u0)| ≤k0ρ(j−n,ω)ek1neλ1µe−λ1Tn(θTj−nω)+2k1 0 X q=1−n eλ1Tq+n(θTj−nω)−λ1Tn(θTj−nω)e−k1qeλ1µ =k0ρ(j−n,ω)ek1neλ1µe−λ1Tn(θTj−nω)+2k1 0 X q=1−n eλ1Tq(θTjω)e−k1qeλ1µ,(4.25) 124 Environmental effects caused by the fractional Brownian motion where ρ(j−n,ω)denotes the radius of the discrete tempered set E(j−n,ω). Now, bytaking limit when ntends to infinity in (4.25), we have lim n→+∞ sup u0∈E(j−n,ω) |Φ(n,j−n,ω,u0)| ≤lim n→+∞ k0ρ(j−n,ω)ek1neλ1µe−λ1Tn(θTj−nω)+2k1 0 X q=−∞ eλ1Tq(θTjω)e−k1qeλ1µ.(4.26) Before analyzingthe addendsin (4.26)in moredetail, wewouldliketonoticethat, thanksto (4.20),for every ε>0there exists qε>0such that for |q| > qε, we have that Tq(θTjω)<q(D−ε). Hence, since λ1is a positive constant, we have λ1Tq(θTjω)−k1qeλ1µ<qhλ1D−k1eλ1µ−λ1εi. Now, weanalyzetheaddendsin(4.26). Ontheonehand,bytaking intoaccount(4.20), for c M>0large enough we obtain that R(j,ω) :=2k1 0 X q=−∞ eλ1Tq(θTjω)e−k1qeλ1µ <2k1 0 X q=−c M eλ1Tq(θTjω)−k1qeλ1µ + −c M−1 X q=−∞ eq[λ1D−k1eλ1µ−λ1ε] ,(4.27) where the last term in (4.27) is well defined thanks to (4.19). Onthe other hand,we analyze the termgivenbythelimitin(4.26). Tothis end,forε>0small enough, ζ:=Dλ1−k1eλ1µ−ελ1>0. Then, thanks to (4.19), (4.20) and the fact that −Tn(θTj−nω)=T−n(θTjω), we have that lim n→+∞ k0ρ(j−n,ω)ek1neλ1µe−λ1Tn(θTj−nω)=lim n→+∞ k0ρ(j−n,ω)ek1neλ1µeλ1T−n(θTjω) ≤lim n→+∞ k0ρ(j−n,ω)ek1neλ1µe−Dnλ1+εnλ1 =k0e−ζjlim n→+∞ ρ(j−n,ω)eζ(j−n)=0. Summing up, from the previous calculations, we deduce that lim n→+∞ sup u0∈E(j−n,ω) |Φ(n,j−n,ω,u0)| ≤ 2k1 0 X q=−∞ eλ1Tq(θTjω)e−k1qeλ1µ=R(j,ω). Havingreachedthispoint,itremainstocheckthattheradiusR(j,ω)givenby(4.21)istempered,which 125 Environmental effects caused by the fractional Brownian motion means that lim j→−∞ eδjR(j,ω)=0. We omit the proof of the temperedness of R(j,ω)since it can be easily made by using similar arguments to the ones shown before. In conclusion, we deduce that the ball centered at the origin with radius given by (4.21) is a tempered discrete absorbing set which will be denotedby ©B(j,ω)ªj∈Z,ω∈Ω. From Theorem 4.3.6, thanks to Proposition B.0.1 in Appendix B, we deduce that there exists a unique discrete random pullback attractor, which will be denoted by {A(i,ω)}i∈Z,ω∈Ω, associated to the discrete dynamical system{Φ(i,j)}i,j∈Zgiven by (4.18), as presented in the following result. Proposition 4.3.1 Thediscretedynamicalsystem(4.18)hasadiscreterandompullbackattractor{A(i,ω)}i∈Z,ω∈Ω given by A(i,ω)=\ n∈N[ m≥n Φ(m,−m+i,ω,B(−m+i,ω)),(4.28) where Bdenotes the absorbing ball obtained in Theorem 4.3.6. Wealsoremarkthatthemeasurabilityofthediscreterandompullbackattractorgivenby(4.28)follows trivially since the stopping times are measurable. Tofinish the section, weprove a resultconcerningtheexistence anduniqueness ofcontinuous random pullbackattractor,whichwillcoincidewith the component{A(0,ω)}ω∈Ωofthe discreteone,associatedto the continuous RDS (4.11). Theorem 4.3.7 There exists a unique continuous random pullback attractor associated to the continuous random dynamical system {ϕ(t,ω)}t≥0,ω∈Ω, which is given by {A(ω)}ω∈Ω, where A(ω)is defined as the component {A(0,ω)}ω∈Ωof the discrete random attractor associated to the discrete random dynamical system (4.18). Proof. The proofof this result is omitted since it is exactly the same that the one in [37]. We would like just tonotethatoneofthemainingredientsistherelationbetweenthecontinuousanddiscretetemperedsets. In fact, if Jbe a continuous tempered set, then EJ(i,ω)=[ −τ∈[ˆ T(θTiω),0] [ u0∈E(θ−τθTiω) {ϕ(τ,θ−τθTiω,u0)} defines discrete tempered set. Moreover, EH(i,ω)=[ τ∈[0,T(θTiω)] [ u0∈E(i,ω) {ϕ(τ,θTiω,u0)} is a discrete tempered set as long as His a discrete tempered set as well. Once taken that into account, the proof basically consists on checking that {A(0,ω)}ω∈Ωis compact, invariant and attracting or, in other 126 Environmental effects caused by the fractional Brownian motion words, every property in Definition B.0.5 in Appendix B holds. 4.3.3 Analysis of stopping times Inthis section, we develop a deeper analysis of thesequence of stopping times given by (4.14). We will firstly state an estimation of the random number of stopping times on the interval (0,µ], which allows us to prove a result concerning the distribution of those stopping times on the whole real line. Thanks to this, wewill deducethat thesequenceofstopping times{Ti(ω)}i∈Zhasnoaccumulationpoints and,therefore, it completely covers the real line. In addition, this last result will guarantee that condition (4.19) in Lemma 4.3.6 can be assumedbytaking small enough noise. Firstly, for H>1/2, weset 1/2 <β<β′<H. Thus, let us consider Ωthe{θt}t∈R−invariantset ofpaths ω:R→Rwhichareβ′−Höldercontinuousonanycompactsubintervalofthereallineandarezeroatzero. Then, Ωis {θt}t∈R−invariant (see [21]). Lemma 4.3.2 Let ω∈Ω. The mapping t7→ |||ω|||β′,0,tis continuous on R+, and the mapping t7→ |||ω|||β′,t,0 is continuous on R−. Proof. We only prove the continuity of the first mapping since the second one can be treated similarly. To this end, if t≥t0, let us define ωt0as follows ωt0(s)=   ω(s), for s<t0, ω(t0), for s≥t0. Thus, for t≥t0we have ¯¯|||ω|||β,0,t− |||ω|||β,0,t0¯¯=¯¯|||ω|||β,0,t− |||ωt0|||β,0,t¯¯≤ |||ω−ωt0|||β,0,t= |||ω|||β,t0,t, whence limsup t→t0¯¯|||ω|||β,0,t− |||ω|||β,0,t0¯¯≤limsup t→t0³|||ω|||β′,t0,t(t−t0)β′−β´ ≤limsup t→t0 |||ω|||β′,0,tlim t→t0 (t−t0)β′−β=0, then, by the same argument, we candeduce that liminf t→t0¯¯|||ω|||β,0,t− |||ω|||β,0,t0¯¯=0. Hence, lim t→t0 |||ω|||β,0,t= |||ω|||β,0,t0. 127 Environmental effects caused by the fractional Brownian motion We note that the case t≤t0can be proved similarly. We would like to remark that the stopping time T(ω)given by (4.13) iswell defined and, in addition, it satisfies T(ω)∈(0,µ]. Asweexplainedinthelastparagraph,webeginprovidinganestimateofthenumberofstoppingtimes on the interval (0,µ]which is stated in the next result. Proposition 4.3.2 Let N(ω)∈Nbe the random number of stopping times in (0,µ].Then, for ω∈Ω, we have N(ω)≤µÃ|||ω|||β′,0,µ+µ1−β′ µ!1 β′ .(4.29) In addition, limsup n→+∞ 1 µn n−1 X i=0 N(θiµω)≤E sup r∈[−µ,0] Ã|||θrω|||β′,0,µ+µ1−β′ µ!1 β′ =:1 µd(4.30) for all ωin a {θt}t∈R−invariant set of full measure where d=d(µ,ω)∈(0,1]. Proof. For every ω∈Ω, since T(ω)<µ, thanks to (4.13) we have µ=sup 0≤s<t≤T(ω) |ω(t)−ω(s)| |t−s|β′|t−s|β′−βT(ω)β+T(ω) ≤ |||ω|||β′,0,T(ω)T(ω)β′−βT(ω)β+T(ω) =T(ω)β′h|||ω|||β′,0,T(ω)+T(ω)1−β′i <T(ω)β′h|||ω|||β′,0,µ+µ1−β′i whence we obtain T(ω)≥õ |||ω|||β′,0,µ+µ1−β′!1 β′ . The same bound can be proved for T(θTi(ω)ω)as long as Ti+1(ω)≤µ, as follows µ= |||θTi(ω)ω|||β,0,T(θTi(ω)ω)T(θTi(ω)ω)β+T(θTi(ω)ω) ≤ |||ω|||β′,Ti(ω),Ti+1(ω)T(θTi(ω)ω)β′−βT(θTi(ω)ω)β+T(θTi(ω)ω) 128 Environmental effects caused by the fractional Brownian motion ≤T(θTi(ω)ω)β′h|||ω|||β′,0,µ+T(θTi(ω)ω)1−β′i, thus, we have T(θTi(ω)ω)≥õ |||ω|||β′,0,µ+µ1−β′!1 β′ . By definition, N(ω)isthelargestnumbersuchthat TN(ω)(ω)≤µ. Then,thanksto thedefinition ofthe stopping times (4.14) we obtain µ≥TN(ω)(ω)= N(ω)−1 X i=0 T(θTi(ω)(ω)) ≥N(ω)õ |||ω|||β′,0,µ+µ1−β′!1 β′ , whence we can easily deduce (4.29). Now, let us note that supr∈[−µ,0] |||θrω|||β′,0,µ≤ |||ω|||β′,−µ,µand the right side of the last equality has finite moments, then (see [58]) we have E sup r∈[−µ,0] Ã|||θrω|||β′,0,µ+µ1−β′ µ!1 β′ < +∞. In fact, since N(θiµω)≤µÃ|||θiµω|||β′,0,µ+µ1−β′ µ!1 β′ ≤Zµ 0 sup r∈[−µ,0] Ã|||θr+q+iµω|||β′,0,µ+µ1−β′ µ!1 β′ dq and thanks to Lemma 4.2.2, we have n−1 X i=0 N(θiµω)≤ n−1 X i=0Zµ 0 sup r∈[−µ,0] Ã|||θr+q+iµω|||β′,0,µ+µ1−β′ µ!1 β′ dq =Zµn 0 sup r∈[−µ,0] Ã|||θr+qω|||β′,0,µ+µ1−β′ µ!1 β′ dq, whence we can deduce (4.30) since limsup n→+∞ 1 µn n−1 X j=0 N(θiµω)≤lim n→+∞ 1 µnZµn 0 sup r∈[−µ,0] Ã|||θr+qω|||β′,0,µ+µ1−β′ µ!1 β′ dq =E sup r∈[−µ,0] Ã|||θrω|||β′,0,µ+µ1−β′ µ!1 β′ . 129 Environmental effects caused by the fractional Brownian motion Thankstotheergodictheoremforcontinuoustime(see[4]),weknowthatthereexistsa{θt}t∈R−invariant set of full measure where this convergence is verified. In addition, we obtain that 1 d:=µE sup r∈[−µ,0] Ã|||θrω|||β′,0,µ+µ1−β′ µ!1 β′ ≥µEõ1−β′ µ!1 β′ =1. Now,westatetwotechnicallemmaswhoseproofscanbefoundin[30]. Hence,weomitthecorresponding proofs. Lemma 4.3.3 For any ω∈Ωand i∈N, let us denote Mi(ω)the random number of stopping times in the interval (iµ,(i+1)µ]. Then, Mi(ω)∈{N(θiµω),N(θiµω)+1}, where N(·)was defined in Proposition 4.3.2. Lemma 4.3.4 Let µ∈(0,1). For any small δ>0such that |||ω|||β,0,µ<δ, there exists a {θt}t∈R−invariant set of full measure Ω′and ¯ d=¯ d(µ,ω)such that, for ω∈Ω′, limsup n→+∞ 1 n n−1 X i=0 1Mj(·)>N(θiµ·)(ω)≤¯ d. Now, we establish the following result which will allow us to make some conclusions concerning the work made through the chapter. Proposition 4.3.3 Under the assumptions of Lemma 4.3.4, we have liminf k→+∞ Tk(ω) k>D, on a {θt}t∈R−invariant set of full measure, where D=D(µ,ω)=µd 1+d¯ d∈(0,µ] such that lim µ→0D(µ,ω)=0, lim ω→0D(µ,ω)=µ.(4.31) Proof. Let Mj(ω)∈Nbe the random number of stopping times in (jµ,(j+1)µ]for some j∈Z. Then, given k∈N, we can take n=n(k,ω)∈Nsuch that n−1 X j=0 Mj(ω)<k≤ n X j=0 Mj(ω). 130 Environmental effects caused by the fractional Brownian motion Then, thanks to the order or the stopping times we know that Tk(ω)>TPn−1 j=0Mj(ω)(ω), whence we obtain liminf k→+∞ Tk(ω) k≥liminf n→+∞ TPn−1 i=0Mi(ω)(ω) Pn i=0Mi(ω)≥liminf n→+∞ (n−1)µ Pn i=0N(θiµω)+Pn i=01Mi(·)>N(θiµ·)(ω), where the last inequality can be deduced from the fact that TPn−1 i=0Mj(ω)(ω)is the last stopping time in ((n−1)µ,nµ]and Lemma 4.3.3. On the other hand, by taking into account (4.30) and Lemma 4.3.4, we have that limsup n→+∞ Pn j=0N(θiµω)+Pn j=01Mj(·)>N(θiµ·)(ω) (n−1)µ=1 µlimsup n→+∞ Pn j=0N(θiµω)+Pn j=01Mj(·)>N(θiµ·)(ω) (n+1)µ ≤1 µd+1 µ ¯ d, holds on a {θt}t∈R−invariant set of full measure. Then, by defining D=µd 1+d¯ d, both properties in (4.31) are verified as long as the contribution of |||ω|||β,0,µis small, since in this case, d→1and ¯ d→0. We would like to remark that, thanks to Proposition 4.3.3, it is possible to consider a small noise such that condition (4.19) is Theorem 4.3.6 is clearly justified. Let us remember that condition (4.19) played an important role when obtaining the radius of the discrete absorbing set which allowed us to conclude the existence and uniqueness of the discrete random pullback attractor and, consequently, the existence anduniquenessofthecontinuousone. Inaddition,Proposition4.3.3alsoguaranteesassumption(4.20) in Theorem 4.3.6 which, moreover, means that there are no accumulation points in the sequence of stopping times given by (4.14). Remark 4.3.2 We note that, as pointed out at the beginning of the chapter, the same analysis could be carried out for the corresponding stochastic chemostat model with wall growth affected by a fractional Brownian motion.To be more precise, we recall that it is also possible to rewrite the model with wall growth in the abstract formulation given by (4.8) such that Properties (P1)-(P3) are held in this case as well. In this way, we could redo the chapter again for the model with wall growth and the same results that the ones provided for the case without wall growth could be proved. 131