Dynamics of some stochastic chemostat models with multiplicative noise

In this paper we study two stochastic chemostat models, with and without wall growth, driven by a white noise. Specifically, we analyze the existence and uniqueness of solutions for these models, as well as the existence of the random attractor associated to the random dynamical system generated by the solution. The analysis will be carried out by means of the well-known Ornstein-Uhlenbeck process, that allows us to transform our stochastic chemostat models into random ones.

1. Introduction. Chemostat refers to a laboratory device used for growing microorganisms in a cultured environment and has been regarded as an idealization of nature to study competition modeling in mathematical biology, which is a really important and interesting problem since they can be used to study recombinant problems in genetically altered microorganisms (see e.g. [17,18]), waste water treatment (see e.g. [13,25]) and play an important role in theoretical ecology (see e.g. [2,12,16,23,29,31,32,34]). Derivation and analysis of chemostat models are well documented in [26,27,33] and references therein.
Two standard assumptions for simple chemostat models are: 1) the availability of the nutrient and its supply rate are fixed and 2) the tendency of microorganisms to adhere to surfaces is not taken into account (see e.g. [7,8]). However, these are very strong restrictions as the real world is non-autonomous and stochastic and this justifies the analysis of stochastic chemostat models, with and without wall growth.
Let us first consider the following chemostat model without wall growth dS dt where S(t) and x(t) denote concentrations of the nutrient and the microbial biomass, respectively; S 0 denotes the volumetric dilution rate, a is the half-saturation constant, D is the dilution rate and m is the maximal consumption rate of the nutrient and also the maximal specific growth rate of microorganisms. We notice that all parameters are supposed to be positive and a function Holling type-II is used as functional response of the microorganisms describing how the nutrient is consumed by the species (see e.g. [28] for more details and biological explanations about this model).
Our aim in this paper is to perturb system (1)-(2) by a noisy input such that the perturbed one becomes a more realistic model of a chemostat. Recently, in [4] the authors have analyzed system (1)-(2) by replacing the dilution rate D by D+αẆ (t), where W (t) is a Wiener process. Even though in that paper the existence and uniqueness of solutions, as well as the existence of the corresponding attractor have been stated, biologically the model does not seem completely realistic, since the substrate S in the corresponding stochastic chemostat model can take negative values. We want to overcome this biological inconsistence by considering a different kind of stochastic perturbation.
We would like to emphasize that one may consider several alternatives to model randomness and stochasticity. We will use a technique based in the one carried out by Fudenberg and Harris in [19] or by Foster and Young in [15], in which the first idea was to consider a stochastic perturbation of the payoff function in continuoustime replicator dynamics. In other words, we could write our model as and then we could add some stochastic perturbation α iẆi to the expected payoff f i (·, ·), for i ∈ {1, 2}, instead of adding it directly to dS/dt and dx/dt, as follows ] , or, equivalently, In this way, the populations S and x will always remain positive for any realization of the Wiener processes W i . In fact, as explained in [19], it can be understood as the payoff to play some strategy i subjected to some external perturbations due to, for example, the weather.
Moreover, in the paper by Imhof and Walcher (see [22]) the authors justify mathematically that it could be reasonable to consider the following stochastic chemostat model where W 1 and W 2 are independent Wiener processes. To this end, a discrete Markov chain is considered for some increment δt and the convergence to the solution of the original stochastic equation is proved as δt tends to zero, whenever it exists a unique solution (see [22] for a more detailed explanation). Motivated by this feature, in this paper we consider a noisy term in each equation (1)- (2) in the same fashion as in [22], which ensures the positivity of both the nutrient and biomass, although does not preserve the washout equilibrium from the deterministic to the stochastic model. More precisely, we consider now the following system, which is understood in the Itô sense where W (t) is a standard Brownian motion, and α ≥ 0 represents the intensity of noise.
We remark that, in order to make the calculations much more tractable and clear, we consider the same noise in both equations, even though a similar analysis could be developed by using different Brownian motions in each equation. This leads to more complicated technicalities that we prefer to avoid in this first approach.
We would also like to note that there are not special reasons to consider the sign minus (−) in front of the stochastic terms, instead of the positive one used in [22], since the choice does not cause any effect over the behavior of our system. Now, by using the well-known conversion between Itô and Stratonovich senses we obtain the following stochastic chemostat without wall growth whereD Before analyzing the previous system, 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 (1)-(2), Caraballo and Han proved in a recently published book (see [6]) the existence of a unique axial equilibrium (S 0 , 0) which is asymptotically stable provided D > m, therefore this situation corresponds to the extinction of the microorganism. However, if D < m and aD/(m − D) < S 0 the axial equilibrium becomes unstable and a unique positive globally asymptotically stable equilibrium appears inside the positive quadrant, i.e., persistence of the microorganism 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 can be deduced related to the asymptotic behavior of the system.
Regarding the stochastic chemostat model (SCM), we prove in this paper that there exists a unique global random attractor which is given by singleton components (S 0 Dρ * (ω), 0) provided D + α 2 /2 > m (see Section 3.1 for more details). Otherwise, the unique global random attractor is contained in a segment whose intersection with the axes S = 0 and x = 0 is reduced to two single points.
In light of the previous facts, observe that when D < m and aD/(m − D) < S 0 we can choose α, large enough, such that D + α 2 /2 > m. This means that persistence of the microorganism holds for (DCM), while for (SCM) we have extinction since the global random attractor becomes the single random point (S 0 Dρ * (ω), 0). This fact is closely related to the stabilizing effects that Itô's noise can produce on deterministic systems. However, if we considered a Stratonovich interpretation for our perturbation at the beginning of our study, then we would have obtained D instead ofD in (3)-(4); in other words, assumption D + α 2 /2 > m in (SCM) would become D > m, the same that we had for (DCM). Consequently, no stabilizing effect is produced by the noise (see [3,6,21] and Remark 3.3 in [24] 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 appropiate choice in a specific application is [30], where the author presents a criterion for determining which interpretation of the noise is the most useful in his work.
Up to now, we have just mentioned the chemostat model without wall growth. Nevertheless, we are also interested in studying the equivalent model with wall growth since it will allow us to work in a more realistic situation and we will also be able to obtain more useful results from the biological point of view. Then, let us now introduce the simplest chemostat model with wall growth where S(t), x 1 (t) and x 2 (t) denote concentrations of the nutrient and the two different microorganisms, respectively; b ∈ (0, 1) describes the fraction of dead biomass which is recycled, ν > 0 is the collective death rate coefficient, r 1 > 0 and r 2 > 0 represent the rates at which the species stick on to and shear off from the walls, respectively, and 0 < c ≤ m is the growth rate coefficient of the consumer species. By introducing again a white noise in each equation of (6)-(8) and using the conversion between Itô and Stratonovich interpretations, we finally obtain the following stochastic system with wall growth The paper is organized as follows: in Section 2 we recall some basic results on random dynamical systems. Then, in Section 3 we analyze both random chemostat models, with and without wall growth, and we provide some results regarding existence and uniqueness of global solution just like the generation of a random dynamical system and existence of random pullback attractor, describing its internal structure explicitly. Moreover, in Section 4 we use a conjugation result in order to explain how the global attractor behaves in the stochastic model. Finally, we state some numerical simulations to illustrate our study and some final comments in Section 5.
2. Random dynamical systems. Although there are very good references (see e.g. [1]) in the literature which provide a very detailed information about random dynamical systems (RDSs), we prefer to recall very briefly here some definitions and results to make our presentation as much self-contained as possible.
is a probability space and the family of mappings θ t : Ω → Ω satisfies Definition 2.2. Let (Ω, F, P) be a probability space. A random set K is a measurable subset of X × Ω with respect to the product σ−algebra B(X) × F. Moreover K will be said a closed or a compact random set if K(ω) = {x : (x, ω) ∈ K}, ω ∈ Ω, is closed or compact for P−almost all ω ∈ Ω, respectively.
In what follows we use E(X) to denote the set of all tempered random sets of X.

Remark 1.
When the state space X = R d as in this paper, the asymptotic compactness follows trivially.
Lemma 2.6. Let φ u be an RDS on X. Suppose that the mapping T : Ω × X → X possesses the following properties: for fixed ω ∈ Ω, T (ω, ·) is a homeomorphism on X, and for x ∈ X, the mappings T (·, x), is a (conjugated) RDS.
3. Random chemostat. In this section we will study the stochastic systems (3)-(4) and (9)-(11) by transforming them into differential equations with random coefficients. Let W be a two sided Wiener process. Kolmogorov's theorem ensures that W has a continuous version, that we will denote by ω, whose canonical interpretation is as follows: let Ω be defined by F the Borel σ−algebra on Ω generated by the compact open topology (see [1] for details) and P the corresponding Wiener measure on F. We consider the Wiener shift flow given by is a metric dynamical system. Now let us introduce the following Ornstein-Uhlenbeck process on (Ω, F, P, which solves the following Langevin equation (see [1,10]) Proposition 3.1 (See [1,10]). There exists a θ t -invariant set Ω ∈ F of Ω of full P measure such that for ω ∈ Ω, we have (i) the random variable |z * (ω)| is tempered.
(ii) the mapping is a stationary solution of (12) with continuous trajectories; (iii) in addition, for any ω ∈Ω: In what follows we will consider the restriction of the Wiener shift θ to the set Ω, and we restrict accordingly the metric dynamical system to this set, that is also a metric dynamical system, see [5]. For simplicity, we will still denote the restricted metric dynamical system by the old symbols (Ω, F, P, {θ t } t∈R ).
From now on, we denote X :

Random chemostat without wall growth.
In what follows we use the Ornstein-Uhlenbeck process to transform (3)-(4) into a random system. To this end, we first define two new variables σ and κ as follows For the sake of simplicity, and when no confusion is possible, we will write z * instead of z * (θ t ω), and σ and κ instead of σ(t) and κ(t).
Hence, it is straightforward that Next we prove that the random chemostat (14)-(15) generates an RDS.
Therefore, thanks to classical results from the theory of ordinary differential equations, system (16)-(17) possesses a unique local solution. Let us check now that in fact this solution is a global one.
We define V (t) := σ(t) + κ(t) and thanks to (16)- (17) we have By solving the previous differential equation we obtain hence V is clearly bounded by above by an expression which does not blow up.
Now we study the existence of a random attractor, describing its internal structure if possible.
First we define ∥ · ∥ as By replacing ω by θ −t ω in (18), we have and therefore sinceD given by (5) is always positive, where ρ * (ω) is defined by Note that the above integrand converges to zero when τ goes to infinity, but not the integral. Moreover, ρ * (ω) has sub-exponential growth. Therefore, for any given ε > 0, there exists T E (ω, ε) > 0 such that Hence, from Proposition 2.1, the RDS generated by the system (16)-(17) possesses a unique random attractor given by A = {A(ω)} ω∈Ω ⊂ B ε (ω) for any ε > 0. Thus A = {A(ω)} ω∈Ω ⊂ B 0 (ω), i.e., we have the following expression for each component of our attractor (5) assume thatD > m. Then, the random attractor A associated to the RDS φ u has the following structure:

Proposition 3.3. ForD defined by
Proof. Thanks to (15) we know that dκ dt ≤ −(D − m + αz * )κ, whose solution, after replacing ω by θ −t ω and making t go to infinity, tends to zero providedD > m, thus the internal structure of the attractor in this case consists of singleton subsets A(ω) = (S 0 Dρ * (ω), 0) which means that there is not persistence of the microorganism (see Figure 2 in Section 5).
However, we cannot ensure the persistence of the microorganism in caseD ≤ m by 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 (see Figure 1 in Section 5).

Random chemostat with wall growth.
In what follows we use the Ornstein-Uhlenbeck process to transform (9)-(11) into a random system. Similarly to Section 3.1, we first define three new variables σ, κ 1 and κ 2 as follows σ(t) = S(t)e αz * (θtω) , κ 1 (t) = x 1 (t)e αz * (θtω) and κ 2 (t) = x 2 (t)e αz * (θtω) . (21) By differentiation, we obtain the following random system Now, we define two new variables (25) in order to transform our random system (22)-(24) into another system which will be more useful to understand the dynamics of the model. For the sake of simplicity we will write κ and ξ instead of κ(t) and ξ(t).
On the other hand, thus we have hence κ does not blow up at any finite time either. As a result, σ(t) = V (t) − κ(t) does not blow up. Therefore, the unique local solution to system (30)-(31) can be extended to a unique global one. Furthermore, therefore we obtain the following inequality, which will be further very useful It is straightforward to verify, similarly to the case without wall growth, that the global solution u(t) of (30)-(31) belongs to X for any initial data u 0 ∈ X and t ∈ R + . Now we can define the mapping φ u : R + × Ω × X → X given by φ u (t, ω)u 0 := u(t; ω, u 0 ), for all t ≥ 0, u 0 ∈ X , ω ∈ Ω.
On the other hand, from (27) we obtain the following inequalities Moreover, we can easily obtain the next lower bound from (26) where c ξ is defined as where ξ * is given by (29). By using (37) we are able to solve (38) whichever the sign of bνc ξ − m, so that we split our analysis into two different cases.
for any ε > 0. In this case the random attractor satisfies In that case we are not able to establish conditions to ensure the persistence of both microorganisms. However, the numerical simulations show that we can obtain persistence in the current case (see Figures 3-4 in Section 5).
• Case B: If bνc ξ − m < 0 holds, we distinguish two cases again: for any ε > 0. In this case the random attractor satisfies which means that there is not persistence of the microorganisms (see Figures 5-6 in Section 5).
In this case the global attractor satisfies A ⊂ B 0 (ω). We are not able to guarantee the persistence of the microorganisms even though the numerical simulations show that we can obtain it (see  in Section 5). Finally, we state Table 1 to summarize the results of the previous study.
does not provide any extra information does not provide any extra information does not provide any extra information Table 1. Internal structure of the random attractor -Random chemostat model with wall growth 4. Existence of the random attractor for the stochastic system.
Moreover, the global random attractor of the random system without wall growth (14)- (15) , the global random attractor of the system (3)-(4), where In other words, each component A T (ω), ω ∈ Ω, of our attractor can be written as ) .
Moreover, we know that the internal structure of the attractor consists of singleton subsets A T (ω) = ( S 0 Dρ * (ω)e −αz * (ω) , 0 ) as long asD > m and we cannot ensure the persistence of the microorganism otherwise. However, our simulations show that we can get the persistence for several values of the parameters (see Figures  1-2 in Section 5).

Stochastic model with wall growth.
We have also proved that the system (30)-(31) has a unique global solution u(t; ω, u 0 ) which remains in X for all u 0 ∈ X and generates the RDS φ u . Now, we define a mapping T : Ω × X −→ X as in Section 4.1.
Moreover, the global random attractor of the random system with wall growth (30)- (31) , the global random attractor of the system (9)-(11), where } . Table 2 in the next page shows information on the random attractor A T = { A T (ω)} ω∈Ω , taking into account the analysis carried out at the end of Section 3.2.
In this way we integrate the equation in (39) on τ j−1 ≤ t ≤ τ j for some arbitrary j ∈ {0, . . . , N } and we use the following approximations of the integrals ∫ τj τj−1 f (X(s))ds ≈ f (X j−1 )δt and ∫ τj τj−1 g(X(s))dW (s) ≈ g(X j−1 )δW j , where δW j := W (τ j ) − W (τ j−1 ) ∼ N (0, δt) are independent normally distributed random variables. Hence, we can already define the following numerical scheme given by X j = X j−1 + f (X j−1 )δt + g(X j−1 )δW j for j = 1, . . . , N and we obtain the simulations below for different values in the parameters of both models without and with wall growth, where the dashed line corresponds to the deterministic solutions and the other lines to the stochastic ones. Firstly, we will show some simulations of the stochastic chemostat model without wall growth. Now, we will show some simulations for the stochastic chemostat model with wall growth by displaying two different panels in each figure: on the left we will show a general point of view of the dynamics; on the right, the viewer is supposed to be looking at the dynamics from point (S 0 , x 01 , 0) in order to make the reader easier check whether the populations involved in our model remain strictly positive or not. Moreover, the thick black asterisk denotes the initial value (S 0 , x 01 , x 02 ).