Random and stochastic disturbances on the input flow in chemostat models with wall growth

RANDOM AND STOCHASTIC DISTURBANCES ON THE INPUT FLOW IN CHEMOSTAT MODELS WITH WALL GROWTH Javier L´ opez-de-la-Cruz∗ Dpto. Ecuaciones Diferenciales y An´alisis Num´erico Facultad de Matem´aticas, Universidad de Sevilla C/ Tarfia s/n, Sevilla 41012, Spain Abstract. In this paper we analyze a chemostat model with wall growth where the input flow is perturbed by two different stochastic processes: the well-known standard Wiener process, which leads into several drawbacks from the biological point of view, and a suitable Orsntein-Uhlenbeck process depending on some parameters which allow us to control the noise to be bounded inside some interval that can be fixed previously by practitioners. Thanks to this last approach, which has already proved to be very realistic when modeling other simplest chemostat models, it will be possible to prove the persistence and coexistence of the species in the model without needing the theory of random dynamical systems and pullback attractors needed when dealing with the Wiener process. This is an advantage since the theoretical framework in this paper is much less complicated and provide us much more information than the other. 1. Introduction The chemostat is a bioreactor very used to study both biological and ecological processes and has many applications in the real life, for instance, that is useful when modeling and analyzing wastewater treatment processes, genetically altered organisms, antibiotic production models and fermentation models of beer and wine, to name some of the most interesting applications. It is worth mentioning that the chemostat has been subjected to a large number of scientific publications and books, not only in biology and ecology but also in mathematics, in fact, there exists a recent area which is call the theory of chemostat where many researchers are involved in the last years. This interest arises due to the fact that the chemostat device can be modeled mathematically in a very simple way which reproduces very well the real process and this is the reason which encourages us to study this model. The simplest chemostat consists of three tanks, the feed bottle, the culture vessel and the collection vessel, which are connected by pumps as in Figure 1. The substrate or nutrient is stored in the feed bottle and supplied to the culture vessel, where the interactions between the substrate and the microorganisms (which are 1991 Mathematics Subject Classification. Primary: 92D25, 34F05, 37H10, 60H30, 46N60; Secondary: 37B55, 60H10. Key words and phrases. Chemostat model, wall growth, Ornstein-Uhlenbeck process, attracting sets, forwards attraction. Partially supported by Junta de Andaluc´ıa under Proyecto de Excelencia P12-FQM-1492. ∗Corresponding author: jlop[email protected]. 1 2 J. L ´ OPEZ-DE-LA-CRUZ also called species) take place. Apart from this, in order to keep the volumen in the culture vessel constant, another flow is pumped from there to the collection vessel. In addition, it is also known that microorganisms tend to adhere to the walls of the culture vessel hence we would have to consider two different species in the culture vessel, the ones which are in the liquid media and those ones sticked on to the walls of the recipient. Feed Bottle Collection Vessel Culture Vessel Figure 1. The chemostat model Then, the resulting mathematical model is given by the following system of differential equations ds dt =D(sin −s)−ms a+sx1−ms a+sx2+bνx1,(1) dx1 dt =−(ν+D)x1+cs a+sx1−r1x1+r2x2,(2) dx2 dt =−νx2+cs a+sx2+r1x1−r2x2,(3) where s,x1,x2denote concentration of the nutrient, the microorganisms in the liquid media and the ones sticked on the walls of the culture vessel, respectively. In addition, b∈(0,1) describes the fraction of dead biomass which is recycled, ν > 0 is the collective death rate coefficient of the microbial biomass representing all the aforementioned factors such as diseases, aging, etc. Apart from that, 0 < c ≤m is the growth rate coefficient of the consumer species. Finally, r1>0 and r2>0 represent the rates at which the species stick on to and shear off from the walls of the culture vessel, respectively. Let us now explain some details about the chemostat model with wall growth. Concerning the equations describing the dynamics of the nutrient or substrate, (1), 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 (2), the term −νx1 reflects 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 terms just refer the quantity of microorganisms which RANDOM CHEMOSTAT MODELS WITH WALL GROWTH 3 stick on to and shear off the walls of the culture vessel. Finally, regarding (3), 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 culture vessel so they cannot be removed to the collection vessel hence we just write −νx2to take into consideration the microbial biomass death as a consequence of the biological process. There are many works in the literature concerning the deterministic chemostat models (see [1,2,3,4,5,6,7]). However, some restrictions need to be assumed in these cases, for instance, every parameter is assumed either to be constant or given by a deterministic function depending on the time in the non-autonomous case and these are very strong restrictions, specially if we think that the real world is in most cases random. In this way, our goal is to consider randomness or stochasticity in the chemostat model with wall growth in order to obtain much more realistic mathematical models and obtain useful information about the long-time behavior of the systems. Nevertheless, there are many different ways of modeling stochasticity or randomness in deterministic models. Concerning the chemostat model, Caraballo et al [8,9] analyze the simplest chemostat model without wall growth by perturbing the input flow by means of the standard Wiener process. As a result, many drawbacks are found, for instance, it is not possible to prove the positiveness of solution of the resulting stochastic chemostat model due to the huge fluctuations (positive or negative) that the white noise could have. In fact, some state variables could take negative values which is completely unrealistic from the point of view of the applications. In addition, it is not possible to ensure the persistence or coexistence of the species in study either, the main goal pursued by biologists, due to the same reason concerning the white noise. In order to save the previous drawbacks, and motivated by the paper of Imhof and Walcher (see [10]), Caraballo et al [11] analyze both chemostat models without and with wall growth by means of the standard Wiener process but, in this case, the way of perturbing the model is quite different. More precisely, the authors directly add the noise in every equation of the differential system such that the positiveness of solution could be easily proved (see [11] for details). Nevertheless, since the standard Wiener process has not bounded variation paths, it is not possible to prove the persistence or coexistence of the species in this case either, which is, needless to say, an important drawback also in this case. Thus, Caraballo et al [12,13] consider a new noise based on the well-known Ornstein-Uhlenbeck (O-U) process depending on some parameters such that every realization of the noise can be ensured to be bounded inside a positive interval. Thanks to this relevant idea, the authors are able to prove the positiveness of global solution of the simplest chemostat model without wall growth and, what is much more interesting, they are able to prove the persistence of the total microbial biomass by proving the existence of absorbing and attracting sets which are deterministic, i.e., they do not depend on the realization of the noise. In addition, the dynamics of the resulting random chemostat is analyzed forwards in time, differently to the case when considering the standard Wiener process where a more 4 J. L ´ OPEZ-DE-LA-CRUZ complicated framework based on the theory of random dynamical systems and pullback attractors is needed. In view of the advantages of using the new suitable Ornstein-Uhlenbeck process presented in [13], in this paper our aim is to use it to perturb the input flow in the chemostat model with wall growth (1)-(3) in the same way that in [13]. As it will be explained after, this allows us to prove the positiveness of solution and we will be able to ensure the persistence of the total microbial biomass and also the coexistence of both species individually, under some conditions on the parameters of the model, in the sense that there exists a number η > 0 such that for any non null initial biomass x1(0) and x2(0) each realization satisfies (4) lim inf t→+∞xi(t)≥η > 0, for i= 1,2, which is stronger than the one used in other many papers lim inf t→+∞x(t)>0 (see, for instance, [10]). Apart from the previous advantages, we also achieve some improvements when comparing this approach with the one by Xu et al in [14] since, even though they consider the chemostat model without wall growth, they need the amplitude of the noise to be small enough in order to ensure the persistence of the microbial biomass (see Theorem 1.2 and Section 4) whereas, in our case, modeling the disturbances by using the O-U process, there is no discussion needed on the amplitude of the noise to ensure the persistence (which is, in addition, in the stronger sense (4)). On the other hand, the authors in [14] prove the results in probability while we prove every results for every realization of the noise in a set of events of full measure. Finally, we will also compare the results obtained with the ones when perturbing the input flow in the chemostat model with wall growth by using the standard Wiener process in order to be able to make a comparison between both ways of modeling. The paper is organized as follows: in Section 2we introduce the suitable Ornstein- Uhlenbeck process which was presented in [13]. In Section 3we analyze the chemostat model with wall growth (1)-(3) where the input flow is perturbed by means of the suitable O-U process. To this end, we prove the existence and uniqueness of global solution which remains inside the positive octant for every positive initial value as well as the existence of a deterministic attracting set whose internal structure will provide us useful information about the long-time behavior of species. Finally, we show some numerical simulations to support the result from the theory. Therefore, in Section 4we briefly present the chemostat model with wall growth where the input flow is perturbed by the standard Wiener process and we also display some numerical simulations. Then, in Section 5we compare both ways of modeling analyzed in the previous sections in order to see the difference and we also display some numerical simulations. Eventually, in Section 6we state some final comments as conclusions of this work. RANDOM CHEMOSTAT MODELS WITH WALL GROWTH 5 2. The Ornstein-Uhlenbeck process In this section, we present the key in our current work which simply consists of perturbing the deterministic models by means of a suitable O-U process defined as the following random variable (5) z∗ β,γ(θtω) = −βγ 0 Z −∞ eβsθtω(s)ds, t ∈R, ω ∈Ω, β, γ > 0, where ωdenotes a standard Wiener process in a certain probability space (Ω,F,P), βand γare positive parameters which will be explained in more detail below and θtdenotes the usual Wiener shift flow given by θtω(·) = ω(·+t)−ω(t), t ∈R. We note that the O-U process (5) can be obtained as the stationary solution of the Langevin equation (6) dz +βzdt =γdω. The O-U process given by (5) is a stationary mean-reverting Gaussian stochastic process where β > 0 is a 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 γ > 0 is a volatility constant which represents the variation or the size of the noise independently of the amount of the noise α > 0. In fact, the O-U process can describe the position of some particle by taking into account the friction, which is the main difference with the standard Wiener process and makes our perturbations to be a better approach to the real ones than the ones obtained when using simply the standard Wiener process. In addition, the O-U process could be understood as a generalization of the standard Wiener process as well in the sense that it would correspond to take β= 0 and γ= 1 in (5). In fact, the O-U also provides a link between the standard Wiener process and no noise at all, as we will see later. Now, we would like to show the relevant effects caused by both parameters βand γon the evolution of realization of the O-U process. Fixed β > 0.Then, the volatility of the process increases when considering larger values of γand the evolution of the process is smoother when taking smaller values of γ, which is completely reasonable due to the fact that γdecides the amount of noise introduced to dz, the term which measures the variation of the process. Henceforth, the process will be subject to suffer much more disturbances when taking a larger value of γ. This behavior can be observed in Figure 2, where we simulate two realizations of the O-U process with β= 1 and we consider γ= 0.1 (blue) and γ= 0.5 (orange). 6 J. L ´ OPEZ-DE-LA-CRUZ t (time) 0 1 2 3 4 5 6 7 8 9 10 z* β,γ(θtω) -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Perturbation Paths γ=0.1 γ=0.5 Figure 2. Effects of the mean reverting constant on the O-U process Fixed γ > 0.In this case the process tends to go further away from the mean value when considering smaller values of βand the attraction of the mean value increases when taking larger values of β. This behavior is completely logical since βhas a huge influence on the drift of the Langevin equation (5). We can observe this behavior in Figure 3, where we simulate two realizations of the O-U process with γ= 0.1 and we take β= 1 (blue) and β= 10 (orange). t (time) 012345678910 z* β,γ(θtω) -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 Perturbation Paths β=1 β=10 Figure 3. Effects of the volatility constant on the O-U process Once presented the O-U process and the effects that its parameters cause on its behavior, we state now some essential properties which will be crucial for our analysis in this paper. Proposition 2.1. There exists a θt-invariant set e Ω∈ F of Ωof full P−measure such that for ω∈e Ωand β, γ > 0, we have (i) the random variable |z∗ β,γ(ω)|is tempered. (ii) the mapping (t, ω)→z∗ β,γ(θtω) = −βγ 0 Z −∞ eβsω(t+s)ds+ω(t) is a stationary solution of (5)with continuous trajectories; (iii) for any ω∈˜ Ωone has lim t→±∞ |z∗ β,γ(θtω)| t= 0; RANDOM CHEMOSTAT MODELS WITH WALL GROWTH 7 lim t→±∞ 1 tZt 0 z∗ β,γ(θsω)ds = 0; lim t→±∞ 1 tZt 0 |z∗ β,γ(θsω)|ds =E[z∗ β,γ]<∞; (iv) finally, for any ω∈e Ω, lim β→∞ z∗ β,γ(θtω) = 0,for all t∈R. The proof of Proposition 2.1 is omitted here. We refer the readers to [15] (Lemma 4.1) for the proof of the last statement and to [16,17] for the proof of the rest. 3. Random chemostat model with wall growth In this section, we are interested in analyzing the chemostat model with wall growth (1)-(3) where the input flow is perturbed by means of the Ornstein-Uhlenbeck process presented in Section 2. We are going to prove the existence and uniqueness of positive global solution of the resulting system as well as the existence of a deterministic attracting set whose internal structure will be also analyzed to obtain detailed information about the long-time behavior of the model. In addition, some conditions on the parameters involved in the system will be provided to ensure the coexistence of the species in the culture vessel. We would also like to note that, for some positive interval, namely (b1, b2), thanks to the last property in Proposition 2.1, for every fixed event ω∈Ω, it is possible to take β > 0 large enough such that D+αz∗ β,δ(θtω)∈(b1, b2) for every t∈R. 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 =D+αz∗ β,δ(θtω)(sin −s)−ms a+sx1−ms a+sx2+bνx1,(7) dx1 dt =−ν+D+αz∗ β,δ(θtω)x1+cs a+sx1−r1x1+r2x2,(8) dx2 dt =−νx2+cs a+sx2+r1x1−r2x2,(9) where z∗ β,ν(θtω) denotes the Ornstein-Uhlenbeck process defined in Section 2and every parameter and state variable was introduced in Section 1. In this section, X={(x, y, z)∈R3:x≥0, y ≥0, z ≥0}will denote the positive cone in the three-dimensional space. Firstly, we will state a result concerning the existence and uniqueness of global solution of the chemostat model with wall growth (7)-(9). Theorem 3.1. For any initial triple v0:= (s0, x10, x20)∈ X, system (7)-(9)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). 8 J. L ´ OPEZ-DE-LA-CRUZ Proof. Let us recall that the random system (7)-(9) can be rewritten as dv dt =L(θtω)v+F(v, θtω), where L(θtω) =       −D+αz∗ β,δ(θtω)−m+bν −m 0−ν+D+αz∗ β,δ(θtω)−r1+c r2 0r1−ν+c−r2       and F:X × [0,+∞)−→ R3is given by F(η, θtω) =          D+αz∗ β,δ(θtω)sin +ma a+η1 η2+ma a+η1 η3 −ca a+η1 η2 −ca a+η1 η3          , where η= (η1, η2, η3)∈ X. Since z∗ β,δ(θtω) is continuous, Lgenerates an evolution system on R3. Moreover, we notice that F(·, θtω)∈ C1X × [0,+∞); R3whence it is locally Lipschitz with respect to (η1, η2, η3)∈ X. Thus, system (7)-(9) possesses a unique local solution. Now, we prove that the unique local solution of system (7)-(9) is defined for any forward time and 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,δ(θtω)> b1>0 for every t∈R,c≤mand b≤1. Thus we have that psatisfies the next random differential inequalities dp dt ≤D+αz∗ β,δ(θtω)sin −b1s−m cb1x1−m cνx2 ≤D+αz∗ β,δ(θtω)sin −ϑhs+m cx1+m cx2i =D+αz∗ β,δ(θtω)sin −ϑp(t), or, in other words, pverifies the following random differential equation (10) dp dt ≤D+αz∗ β,δ(θtω)sin −ϑp(t), where ϑ:= min{b1, ν}>0. By (10), we have (11) p(t; 0, ω, p0)≤p0e−ϑt +sin Zt 0D+αz∗ β,δ(θsω)e−ϑ(t−s)ds. We remark that the integrand in (11) converges to zero for every t≥s≥0 when tgoes to infinity, but not the integral. Moreover, the integral has subexponential growth. RANDOM CHEMOSTAT MODELS WITH WALL GROWTH 9 Therefore, pdoes not blow up at any finite time, thus s,x1and x2do not blow up at any finite time either. Hence, the solution of system (7)-(9) is defined for any forward time, whence we can straightforwardly deduce that the unique local solution of our system (7)-(9) is, in fact, a unique global one. Now, we are going to prove that the previous unique global solution remains in the positive cone Xfor every initial value v0∈ X. To this end, we firstly consider x1≥0 and x2≥0 and we evaluate the random differential equation for the substrate when s= 0 such that we have ds dt s=0 =D+αz∗ β,δ(θtω)sin +bνx1>0 due to the fact that the perturbed input flow is always positive. Moreover, for every s≥0 and x2≥0, from the equation of the microorganisms in the liquid media, we have dx1 dt x1=0 =r2x2≥0 and, for every s≥0 and 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 (7)-(9) remains in the positive cone Xfor every initial value v0∈ X.  3.2. Existence of a deterministic attracting set. In this section, we study the existence of a deterministic compact absorbing set as well as the existence of a deterministic attracting set, both of them forwards in time, for the solutions of our random chemostat model with wall growth (7)-(9). Theorem 3.2. For any given ε > 0, there exists a deterministic compact absorbing set Bε⊂ X for the solutions of system (7)-(9), i.e., there exists some time 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 (11) we obtain p(t; 0, ω, p0)≤p0e−ϑt +sin Zt 0D+αz∗ β,δ(θsω)e−ϑ(t−s)ds ≤p0e−ϑt +sin Zt 0 b2e−ϑ(t−s)ds =p0e−ϑt +sinb2 ϑ1−e−ϑt,(12) since D+αz∗ β,δ(θsω)≤b2for every s∈R. 16 J. L ´ OPEZ-DE-LA-CRUZ for every v0∈F, any ε > 0 and for all t≥TF(ω, ε). Now, we study the differential equation (39) when x=ex, where exis defined as (40) ex= z∗ l−(ν+b2)a+z∗ u c m+c. Then, from (39) 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, any ε∈(0, cex) and for all t≥TF(ω, ε). Hence, as long as (35) is fulfilled, we have that, for any ε∈(0, cex), where exis given by (40), for every v0∈F, there exists some time TF(ω, ε)>0 such that dx(t; 0, ω, x0) dt x=ex >0(41) for all t≥TF(ω, ε). Therefore, from (41) we conclude that, as long as (35) is fulfilled, we have the following lower deterministic 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 =D+αz∗ β,δ(θtω)(sin −s)−ms a+sx+bνξx for every t≥0, whence we obtain that ds(t; 0, ω, s0) dt =D+αz∗ β,δ(θtω)sin −D+αz∗ β,δ(θtω)s(t; 0, ω, s0) −ms(t; 0, ω, s0) a+s(t; 0, ω, s0)x(t; 0, ω, x0) + bνξ(t; 0, ω, ξ0)x(t; 0, ω, x0), for all t≥0 and every initial value v0∈F. Moreover, from (31) and (37), we know that, for each v0∈Fand any ε > 0, there exists some time TF(ω, ε)>0 such 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 (42) 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, RANDOM CHEMOSTAT MODELS WITH WALL GROWTH 17 for every v0∈F, any ε > 0 and for all t≥TF(ω, ε). Now, we study the differential equation (42) when s=es, where esis defined as (42) es=b1sin b2+ 2z∗ u a . Then, from (42) 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, any ε∈(0, z∗ u) and for all t≥TF(ω, ε). Hence, we have that, for any ε∈(0, z∗ u) and every v0∈F, there exists some time TF(ω, ε)>0 such 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, 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 (40), for every given v0∈F, there exists some time TF(ω, ε)>0 such that (43) x(t; 0, ω, x0)>ex and (44) s(t; 0, ω, s0)>es hold true for all t≥TF(ω, ε). As a result, we can deduce that, for any ε∈(0,min{cex, z∗ u}), (45) b B(s,x) ε=(s, x)∈R2 +:x≥ex, s ≥es, z∗ l−ε≤cs +mx ≤z∗ u+ε, where exand esare defined by (40) and (42), respectively, is a deterministic compact absorbing set (forwards in time) for the solutions of system (16)-(17).  It is worth mentioning that we already proved the dynamics of system (16)-(17) to remain inside B(s,x) ε, defined as in (32), forwards in time. In the previous result, as long as (35) is fulfilled, we obtain in addition a smaller deterministic compact absorbing set e B(s,x) εforwards in time as well, defined by (45), which besides is strictly contained in the first quadrant of the two-dimensional space. This fact will be the main key to guarantee the persistence of both species, individually, in the strong 18 J. L ´ OPEZ-DE-LA-CRUZ sense (4). s x ex es z∗ u m z∗ l m z∗ l c z∗ u c Figure 5. Attracting set e B(s,x) 0 Therefore, e B(s,x) 0:= (s, x)∈R2 +:x≥ex, s ≥es, z∗ l≤cs +mx ≤z∗ u is a deterministic attracting set (forwards in time) for the solutions of system (16)- (17), see Figure 5. Remark 1. It is not difficult to check that both ex < z∗ l mand es < z∗ l care satisfied. Finally, we will analyze the dynamics of both species, x1and x2, individually, to prove that both of them also persist as long as (35) holds true. To this end, thanks to (43), (44) 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 (35) holds true. Remark 2. It is possible to improve the deterministic lower bounds obtained in (43) and (44) 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 have that x > exn, instead of x > exas in (43), where exnis given by exn:= z∗ l−(ν+b2)a+z∗ u c m+c n , RANDOM CHEMOSTAT MODELS WITH WALL GROWTH 19 which clearly satisfies exn:= z∗ l−(ν+b2)a+z∗ u c m+c n > z∗ l−(ν+b2)a+z∗ u c m+c=: ex > 0, for any n∈N, since n≥1. 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 (44), 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. 3.4. Numerical simulations. In this section we show several numerical simulations for the random chemostat model with wall growth (7)-(9) in order to support the results provided through this section. The blue dashed lines represent the solution of the corresponding deterministic chemostat model whereas the other ones represent different realizations of the solution of the random system. Moreover, we display four different panels in each figure: there is a big one on the left-hand side showing the general dynamics of the model and there are three smaller panels on the right-hand side where the individual dynamics of the substrate and both species will be presented. On the one hand, we consider in Figure 6sin = 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 and we will take s0= 2.5, x10 = 2, x20 = 2 as initial values. As a result, we have persistence. Figure 6. Persistence of the species in the random chemostat model with wall growth. 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 20 J. L ´ OPEZ-DE-LA-CRUZ 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 7we will increase the quantity of noise to α= 2, the mean reversion constant to β= 4 and the volatility constant to ν= 0.7, respect to the parameters used in the last figure. Figure 7. Persistence of the species in the random chemostat model with wall growth (ii). sin = 4, D= 2, a= 1.6, m= 2, b= 0.5, ν= 1.2, c= 3, r1= 0.2, r2= 0.4, α= 2, β= 4, ν= 0.7 On another hand, in Figure 8we 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 = 2 as initial values for the substrate and both species, respectively. Then, both species become extinct, as can be 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. RANDOM CHEMOSTAT MODELS WITH WALL GROWTH 21 Figure 8. Extinction of the species in the random chemostat model with wall growth. 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 Finally, in Figure 9we 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 8. In this case we can also see easily that both species become extinct, which is not surprising by taking into account what happened in the last figure and the new values of the parameters involved in the disturbances. Figure 9. Extinction of the species in the random chemostat model with wall growth (ii). 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= 2, β= 4, ν= 0.7 22 J. L ´ OPEZ-DE-LA-CRUZ 4. Stochastic chemostat model with wall growth In this section we analyze the chemostat model with wall growth (1)-(3) where the input flow is influenced by a standard Wiener process. Our main goal now is to study the effects produced by a non-bounded noise on a chemostat model with wall growth. Let us recall the deterministic chemostat model with wall growth, which is given by the following differential system ds dt =D(sin −s)−ms a+sx1−ms a+sx2+bνx1,(46) dx1 dt =−(ν+D)x1+cs a+sx1−r1x1+r2x2,(47) dx2 dt =−νx2+cs a+sx2+r1x1−r2x2,(48) where s,x1and x2denote concentrations of the nutrient and the two different microorganisms, respectively, and the parameters were introduced in Section 1. Our aim now is to perturb the input flow by the standard Wiener process in the same way that in the previous section, i.e., we replace the parameter Dby D+α˙ω(t) in (46)-(48), where α > 0 represents the intensity of the noise and ωdenotes the white noise, such that the following stochastic system understood in It¯o sense is obtained ds =D(sin −s)−ms a+sx1−ms a+sx2+bνx1dt +α(sin −s)dω(t), dx1=−(ν+D)x1+cs a+sx1−r1x1+r2x2dt −αx1dω(t), dx2=−νx2+cs a+sx2+r1x1−r2x2dt. After making use of the well-known conversion between It¯o and Stratonovich sense, we have the following stochastic system in Stratonovich sense ds =¯ D(sin −s)−ms a+sx1−ms a+sx2+bνx1dt +α(sin −s)◦dω(t),(49) dx1=−(ν+¯ D)x1+cs a+sx1−r1x1+r2x2dt −αx1◦dω(t),(50) dx2=−νx2+cs a+sx2+r1x1−r2x2dt,(51) where ¯ D=D+α2 2. 4.1. Stochastic chemostat becomes a random chemostat. In this section, we analyze the stochastic chemostat model with wall growth (49)-(51) by performing the variable change σ(t) = (s(t)−sin)eαz∗(θtω),(52) RANDOM CHEMOSTAT MODELS WITH WALL GROWTH 23 κ1(t) = x1(t)eαz∗(θtω),(53) κ2(t) = x2(t),(54) where z∗(θtω) denotes the O-U process z∗ β,1(θtω) defined in Section 2. The last variable κ2remains as x2due to the fact that equation (51) is not affected by the stochastic perturbation. For the sake of simplicity, we will write again z∗instead of z∗(θtω) and σ,κ1,κ2in place of σ(t), κ1(t), κ2(t). From (52)-(54), by differentiation, we obtain the following differential system satisfied by σ,κ1and κ2, respectively, dσ dt =−(¯ D+αz∗)σ− m(sin +σe−αz∗) a+sin +σe−αz∗κ1− m(sin +σe−αz∗) a+sin +σe−αz∗κ2eαz∗ +bνκ1,(55) dκ1 dt =−(ν+¯ D+αz∗)κ1+c(sin +σe−αz∗) a+sin +σe−αz∗κ1−r1κ1+r2κ2eαz∗ ,(56) dκ2 dt =−νκ2+c(sin +σe−αz∗) a+sin +σe−αz∗κ2+r1κ1e−αz∗ −r2κ2.(57) Now, we perform another variable change to transform the random system (55)- (57) into another one where the total biomass and the proportion of one of the species play an important and helpful role. To this end, we define the new state variables κ(t) = κ1(t) + κ2(t),(58) ξ(t) = κ1(t) κ1(t) + κ2(t).(59) We also write in this case κand ξinstead of κ(t) and ξ(t) in order to make the readability easier. From (58)-(59), by differentiation, we have the following random system satisfied by σ,κand ξ, respectively, dσ dt =−(¯ D+αz∗)σ− m(sin +σe−αz∗) a+sin +σe−αz∗ξκ − m(sin +σe−αz∗) a+sin +σe−αz∗eαz∗ κ(1 −ξ) + bνξκ, dκ dt =−νκ −(¯ D+αz∗)κξ +c(sin +σe−αz∗) a+sin +σe−αz∗κ+r1κξ(e−αz∗ −1) + r2(1 −ξ)κ(eαz∗ −1), dξ dt =−(¯ D+αz∗)ξ(1 −ξ)−r1ξ+r2eαz∗ (1 −ξ)−r1(e−αz∗ −1)ξ2 −r2(eαz∗ −1)ξ(1 −ξ). Instead of analyzing now the existence and uniqueness of global solution of previous random system, we assume that there exists a unique global solution of the random chemostat model with wall growth. Particularly, from the random differential equations describing the dynamics of the substrate, we have the following equalities dσ dt σ=0 =−msin a+sin ξκ −msin a+sin eαz∗κ(1 −ξ) + bνξκ, =−msin a+sin κhξ+eαz∗(1 −ξ)i 24 J. L ´ OPEZ-DE-LA-CRUZ =κ−msin a+sin ξ+eαz∗(1 −ξ)+bνξ. Thus, σwill remain positive as long as msin a+sin ξ(t) + eαz∗(θtω)(1 −ξ(t))≤bνξ(t) holds true for every t≥0 and any ω∈Ω or, equivalently, (60) eαz∗(θtω)≤bνξ(t)(a+sin) msin −ξ(t)1 1−ξ(t). Thanks to (59), 0 ≤ξ(t)≤1 for every t≥0 since it is defined as the proportion of microorganisms in the liquid media. Hence, we have to distinguish the following situations: Case 1.- ξtends to one. In this case, the quotient 1/(1 −ξ(t)) tends to infinity, thus (60) could be true. Nevertheless, it would mean that κ2(or its corresponding state variable x2) tends to zero. Consequently, the microbial biomass sticked on the walls of the culture vessel would become extinct, then we would recover the stochastic chemostat model without wall growth analyzed by Caraballo et al in [8,9]. Case 2.- ξdoes not tend to one. In this case, the right-hand side in (60) is bounded for every t≥0 and any ω∈Ω, whereas z∗could take arbitrary large values which means that (60) does not hold true and then σcan take negative values. In addition, it is not difficult to prove the following lower bound for σ, which is given by (61) σ(t)>−(a+sin)eαz∗(θtω), similarly to the proof made in [9] (Theorem ), that implies (62) s(t)>−a. As explained in [8,9], it is not realistic at all from the biological point of view and this is the main reason because we will not develop a deeper analysis of this kind of noise on the chemostat model with wall growth. 4.2. Numerical simulations and final comments. In this section we show some numerical simulations to remark the drawbacks found when using the Wiener process to perturb the input flow in the chemostat model with wall growth. In each figure some different panels are displayed. On the one hand, there is a big panel on the left, where we can see the phase plane showing the dynamics of the stochastic chemostat model with wall growth in Stratonovich sense (49)- (51). On the other hand, three different panels are stated on the right to describe the dynamics of the substrate and both species individually, depending on the time. The blue dashed lines in the big panel represent the solutions of the deterministic system (46)-(48). In addition, we set sin = 1, a= 0.6, m= 3, r1= 0.6, r2= 0.4 and we consider (s(0), x1(0), x2(0)) = (5,2.5,2.5) as initial data. In this way, we present different cases where the value of the rest of the parameters change and the intensity of the noise increases or decreases, in order to show the effect of each one on the dynamics of our model. RANDOM CHEMOSTAT MODELS WITH WALL GROWTH 25 On the one hand, in Figure 10 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 10. Extinction of both species. α= 0.5 Similarly, in Figure 11 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 totally unrealistic from the biological point of view, as explained previously.