Some aspects concerning the dynamics of stochastic chemostats

Some aspects concerning the dynamics of stochastic chemostats Tom´ as Caraballo, Mar´ ıa J. Garrido-Atienza and Javier L´ opez-de-la-Cruz Abstract In this paper we study a simple chemostat model influenced by white noise which makes this kind of models more realistic. We use the theory of random attractors and, to that end, we first perform a change of variable using the Ornstein- Uhlenbeck process, transforming our stochastic model into a system of differential equations with random coefficients. After proving that this random system possesses a unique solution for any initial value, we analyze the existence of random attractors. Finally we illustrate our results with some numerical simulations. 1 Introduction Modeling chemostats is a really interesting and important problem with special interest in mathematical biology, since they can be used to study recombinant problems in genetically altered microorganisms [13, 14], waste water treatment [10, 18] and play an important role in theoretical ecology [2, 9, 12, 17, 22, 23, 24, 26]. Derivation and analysis of chemostat models are well documented in [19, 20, 25] and references therein. Two standard assumptions for simple chemostat models are as follows: (1) the availability of the nutrient and its supply rate are fixed and (2) the tendency of the Tom´ as Caraballo Dpto. Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain. e-mail: [email protected] Mar´ ıa J. Garrido Atienza Dpto. Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain. e-mail: [email protected] Javier L´ opez de la Cruz Dpto. Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain. e-mail: [email protected] 1 2 T. Caraballo, M.J. Garrido-Atienza, J. L´ opez-de-la-Cruz microorganisms to adhere to surfaces is not taken into account. However, these are very strong restrictions as the real world is non-autonomous and stochastic, and this justifies the analysis of stochastic chemostat models. Let us first consider one of the simplest chemostat models, dS dt = (S0−S)D−mSx a+S,(1) dx dt =xmS a+S−D,(2) where S(t)and x(t)denote concentrations of the nutrient and the microbial biomass, respectively; S0denotes 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 as functional response of the microorganism describing how the nutrient is consumed by the species (see [21] for more details and biological explanations about this model). However, we can consider a more realistic model by introducing a white noise in one of the parameters, therefore we replace the dilution rate Dby D+α˙ W(t), where W(t)is a white noise, i.e., is a Brownian motion, and α≥0 represents the intensity of noise. Then, system (1)-(2) is replaced by the following system of stochastic differential equations dS =(S0−S)D−mSx a+Sdt +α(S0−S)dW(t),(3) dx =xmS a+S−Ddt −αxdW(t).(4) System (3)-(4) has been analyzed in [27] by using the classic 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 in [27], our aim in this paper 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 [27]. In addition, we will provide some results which hold with probability one while those from [27] are said to hold in probability. System (3)-(4) is understood in the Itˆ o sense. Then we first consider its equivalent Stratonovich formulation which is given by dS =(S0−S)D+α2 2−mSx a+Sdt +α(S0−S)◦dW(t),(5) Stochastic chemostats 3 dx =xmS a+S−D+α2 2dt −αx◦dW(t).(6) In Section 2 we recall some basic results on random dynamical systems. In Section 3 we start with the study of equilibria and we prove a result related to the existence and uniqueness of global solution of (5)-(6), by using the so-called Ornstein-Uhlenbeck process. Then, we define a random dynamical system and prove the existence of a random attractor for system (5)-(6) giving an explicit expression for it. Finally, in Section 3.5 we show some numerical simulations with different values of αand we can see what happens when αincreases. 2 Random dynamical systems In this section we present some basic results related to random dynamical systems (RDSs) and random attractors which will be necessary for our analysis. For more detailed information about RDSs and their importance, see [1]. Let (X,k·kX)be a separable Banach space and let (Ω,F,P)be a probability space where Fis the σ−algebra of measurable subsets of Ω(called “events”) and Pis the probability measure. To connect the state ωin the probability space Ωat time 0 with its state after a time of telapses, we define a flow θ={θt}t∈Ron Ω with each θtbeing a mapping θt:Ω→Ωthat satisfies (1) θ0=IdΩ, (2) θs◦θt=θs+tfor all s,t∈R, (3) the mapping (t,ω)7→ θtωis measurable, (4) the probability measure Pis preserved by θt, i.e., θtP=P. This set-up establishes a time-dependent family θthat tracks the noise, and (Ω,F,P,θ) is called a metric dynamical system [1]. Definition 1. A stochastic process {ϕ(t,ω)}t≥0,ω∈Ωis said to be a continuous RDS over (Ω,F,P,{θt}t∈R)with state space Xif ϕ:[0,+∞)×Ω×X→Xis (B[0,+∞)×F×B(X),B(X))- measurable, and for each ω∈Ω, (i) the mapping ϕ(t,ω):X→X,x7→ ϕ(t,ω)xis continuous for every t≥0, (ii) ϕ(0,ω)is the identity operator on X, (iii) (cocycle property) ϕ(t+s,ω) = ϕ(t,θsω)ϕ(s,ω)for all s,t≥0. Definition 2. Let (Ω,F,P)be a probability space. A random set Kis a measurable subset of X×Ωwith respect to the product σ−algebra B(X)×F. The ω−section of a random set Kis defined by K(ω) = {x:(x,ω)∈K},ω∈Ω. 4 T. Caraballo, M.J. Garrido-Atienza, J. L´ opez-de-la-Cruz In the case that a set K⊂X×Ωhas closed or compact ω−sections it is a random set as soon as the mapping ω7→ d(x,K(ω)) is measurable (from Ωto [0,∞)) for every x∈X, see [8]. Then Kwill be said to be a closed or a compact, respectively, random set. It will be assumed that closed random sets satisfy K(ω)6=/0 for all or at least for P−almost all ω∈Ω. Remark 1. It should be noted that in the literature very often random sets are defined provided that ω7→ d(x,K(ω)) is measurable for every x∈X. Obviously this is satisfied, for instance, when K(ω) = Nfor all ω, where Nis some non-measurable subset of X, and also when K= (U×F)∪(U×Fc)for some open set U⊂Xand F/∈F. In both cases ω7→ d(x,K(ω)) is constant, hence measurable, for every x∈X. However, both cases give K⊂X×Ωwhich is not an element of the product σ−algebra B(X)×F. Definition 3. A bounded random set K(ω)⊂Xis said to be tempered with respect to {θt}t∈Rif for a.e. ω∈Ω, lim t→∞e−βtsup x∈K(θ−tω) kxkX=0,for all β>0; a random variable ω7→ r(ω)∈Ris said to be tempered with respect to {θt}t∈Rif for a.e. ω∈Ω, lim t→∞e−βtsup t∈R |r(θ−tω)|=0,for all β>0. In what follows we use D(X)to denote the set of all tempered random sets of X. Definition 4. A random set B(ω)⊂Xis called a random absorbing set in D(X)if for any D∈D(X)and a.e. ω∈Ω, there exists TD(ω)>0 such that ϕ(t,θ−tω)D(θ−tω)⊂B(ω),∀t≥TD(ω). Definition 5. Let {ϕ(t,ω)}t≥0,ω∈Ωbe an RDS over (Ω,F,P,{θt}t∈R)with state space Xand let A(ω)(⊂X)be a random set. Then A={A(ω)}ω∈Ωis called a global random D−attractor (or pullback D−attractor) for {ϕ(t,ω)}t≥0,ω∈Ωif (i) (compactness) A(ω)is a compact set of Xfor any ω∈Ω; (ii) (invariance) for any ω∈Ωand all t≥0, it holds ϕ(t,ω)A(ω) = A(θtω); (iii) (attracting property) for any D∈D(X)and a.e. ω∈Ω, lim t→∞distX(ϕ(t,θ−tω)D(θ−tω),A(ω)) = 0, where distX(G,H) = sup g∈G inf h∈Hkg−hkX Stochastic chemostats 5 is the Hausdorff semi-metric for G,H⊆X. Proposition 1. [6, 11] Let B ∈D(X)be a closed absorbing set for the continuous random dynamical system {ϕ(t,ω)}t≥0,ω∈Ωthat satisfies the asymptotic compactness condition for a.e.ω∈Ω, i.e., each sequence xn∈ϕ(tn,θ−tnω)B(θ−tnω)has a convergent subsequence in X when tn→∞. Then ϕhas a unique global random attractor A={A(ω)}ω∈Ωwith component subsets A(ω) = \ τ≥TB(ω)[ t≥τ ϕ(t,θ−tω)B(θ−tω). If the pullback absorbing set is positively invariant, i.e., ϕ(t,ω)B(ω)⊂B(θtω)for all t ≥0, then A(ω) = \ t≥0 ϕ(t,θ−tω)B(θ−tω). Remark 2. When the state space X=Rdas in this paper, the asymptotic compactness follows trivially. Note that the random attractor is path-wise attracting in the pullback sense, but does not need to be path-wise attracting in the forward sense, although it is forward attracting in probability, due to some possible large deviations, see e.g. [1]. The next result ensures when two random dynamical systems are conjugated (see also [3, 4]). Lemma 1. Let ϕube a random dynamical system 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), T −1(·,x)are measurable. Then the mapping (t,ω,x)→ϕv(t,ω)x:=T−1(θtω,ϕu(t,ω)T(ω,x)) is a (conjugated) random dynamical system. 3 Random chemostat In this section we will investigate the stochastic system (5)-(6). To this end, we first transform it into differential equations with random coefficients and without white noise. Let Wbe a two sided Wiener process. Kolmogorov’s theorem ensures that Whas a continuous version, that we will denote by ω, whose canonical interpretation is as follows: let Ωbe defined by Ω={ω∈C(R,R):ω(0) = 0}=C0(R,R), 6 T. Caraballo, M.J. Garrido-Atienza, J. L´ opez-de-la-Cruz Fbe the Borel σ−algebra on Ωgenerated by the compact open topology (see [1] for details) and Pthe corresponding Wiener measure on F. We consider the Wiener shift flow given by θtω(·) = ω(·+t)−ω(t),t∈R, then (Ω,F,P,{θt}t∈R)is a metric dynamical system. Now let us introduce the following Ornstein-Uhlenbeck process on (Ω,F,P,{θt}t∈R) z∗(θtω) = − 0 Z −∞ esθtω(s)ds,t∈R,ω∈Ω, which solves the following Langevin equation [1, 5] dz +zdt =dω(t),t∈R. Proposition 2. ([1, 5]) There exists a θt-invariant set e Ω∈Fof Ωof full Pmeasure such that for ω∈e Ω,we have (i) the random variable |z∗(ω)|is tempered. (ii) the mapping (t,ω)→z∗(θtω) = − 0 Z −∞ esω(t+s)ds+ω(t) is a stationary solution of (7) with continuous trajectories; (iii) in addition, for any ω∈˜ Ω: lim t→±∞ |z∗(θtω)| t=0; lim t→±∞ 1 tZt 0 z∗(θsω)ds =0; lim t→±∞ 1 tZt 0 |z∗(θsω)|ds =E[z∗]<∞. In what follows we will consider the restriction of the Wiener shift θto the set ˜ Ω, and we restrict accordingly the metric dynamical system to this set, that is also a metric dynamical system, see [4]. For simplicity, we will still denote the restricted metric dynamical system by the old symbols (Ω,F,P,{θt}t∈R). Stochastic chemostats 7 3.1 Stochastic chemostat becomes a random chemostat In what follows we use the Ornstein-Uhlenbeck process to transform (5)-(6) into a random system. Let us note that analyzing the equilibria we obtain that the only one is the axial equilibrium (S0,0)and then we define two new variables σand κby σ(t)=(S(t)−S0)eαz∗(θtω),(7) κ(t) = x(t)eαz∗(θtω).(8) For the sake of simplicity we will write z∗instead of z∗(θtω), and σand κinstead of σ(t)and κ(t). On the one hand, by differentiation, we have dσ=eαz∗dS +(S−S0)eαz∗αdz∗ =(S0−S)D+α2 2−mSx a+Sdt +α(S0−S)◦dW(t)eαz∗ +(S−S0)eαz∗α{−z∗dt +dW(t)} = (S0−S)D+α2 2eαz∗dt −mSx a+Seαz∗dt +α(S0−S)eαz∗◦dW(t) −(S−S0)αeαz∗z∗dt +(S−S0)eαz∗α◦dW (t) =−D+α2 2σ−mSκ a+S−ασz∗dt ="−D+α2 2σ−m(S0+σe−αz∗) a+S0+σe−αz∗κ−ασz∗#dt. On the other hand, dκ=eαz∗dx +xeαz∗αdz∗ =xmS a+S−D+α2 2dt −αx◦dW(t)eαz∗+αxeαz∗[−z∗dt +dW(t)] =xmS a+Seαz∗dt +x−D+α2 2eαz∗dt −αxeαz∗◦dW(t) −αxz∗eαz∗dt +αxeαz∗◦dW(t) 8 T. Caraballo, M.J. Garrido-Atienza, J. L´ opez-de-la-Cruz ="m(S0+σe−αz∗) a+S0+σe−αz∗κ−D−α2 2κ−αz∗κ#dt. Thus, we have obtained the following random system dσ dt =−(¯ D+αz∗)σ−m(S0+σe−αz∗) a+S0+σe−αz∗κ,(9) dκ dt =−(e D+αz∗)κ+m(S0+σe−αz∗) a+S0+σe−αz∗κ,(10) where ¯ D:=D+α2 2and e D:=D−α2 2. 3.2 Random chemostat generates an RDS Next we prove that the random chemostat system (9)-(10) generates an RDS. From now on, we denote X:={(x,y)∈R2:x∈R,y≥0}, the upper half-plane. Lemma 2. Assume that D≥α2 2,˜ λ:=˜ Da m−˜ D≥S0.(11) Then for any ω∈Ωand any initial value u0:= (σ0,κ0)∈X, where σ0:=σ(0) and κ0:=κ(0), system (9)-(10) possesses a unique global solution u(·;ω,u0):= (σ(·;ω,u0),κ(·;ω,u0)) ∈C1([0,+∞),X)with u(0;ω,u0) = u0. Moreover the solution mapping generates a random dynamical system ϕu:R+×Ω×X→X defined as ϕu(t,ω)u0=u(t;ω,u0),∀t∈R+,u0∈X,ω∈Ω. Proof. Observe that we can rewrite one of the terms in the previous equations as m(S0+σe−αz∗) a+S0+σe−αz∗κ=m(S0+σe−αz∗+a−a) a+S0+σe−αz∗κ=mκ−maκ a+S0+σe−αz∗ and therefore system (9)-(10) turns into dσ dt =−(¯ D+αz∗)σ−mκ+ma a+S0+σe−αz∗κ,(12) dκ dt =−(e D+αz∗)κ+mκ−ma a+S0+σe−αz∗κ.(13) Stochastic chemostats 9 Denoting u(·;ω,u0):= (σ(·;ω,u0),κ(·;ω,u0)), system (12)-(13) can be rewritten as du dt =L(θtω)·u+F(u,θtω), where L(θtω) = −(¯ D+αz∗)−m 0−(e D+αz∗)+ m and F:X×[0,+∞)−→ R2is given by F(ξ,θtω) =    ma a+S0+ξ1e−αz∗ξ2 −ma a+S0+ξ1e−αz∗ξ2  , where ξ= (ξ1,ξ2)∈X. Since z∗(θtω)is continuous, Lgenerates an evolution system on R2. Moreover, we notice that ∂ ∂ξ2±am a+S0+ξ1e−αz∗ξ2=±am a+S0+ξ1e−αz∗ and ∂ ∂ξ1±am a+S0+ξ1e−αz∗ξ2=∓ame−αz∗ (a+S0+ξ1e−αz∗)2ξ2 so F(·,θtω)∈C(X×[0,+∞);R2)and is continuously differentiable with respect to the variables (ξ1,ξ2), 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 (12)-(13) possesses a unique local solution. Let us check now that in fact this solution is a global one. In order to do that, we split our analysis into two different cases: first, we assume σ(t)≥0 for all t≥0. Thus, from (9)-(10) d dt (σ+κ) = −¯ Dσ−αz∗σ−e Dκ−αz∗κ 16 T. Caraballo, M.J. Garrido-Atienza, J. L´ opez-de-la-Cruz 3.4 Existence of the random attractor for the stochastic chemostat system We have proved that the system (9)-(10) has a unique global solution u(t;ω,u0) which remains in Xfor all u0∈Xand generates the RDS ϕu. Now, we define a mapping T:Ω×X−→ X as follows T(ω,ζ) = T(ω,(ζ1,ζ2)) = T1(ω,ζ1) T2(ω,ζ2)=(ζ1−S0)eαz∗(ω) ζ2eαz∗(ω) whose inverse is given by T−1(ω,ζ) = S0+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 (7)-(8). Since Tis a homeomorphism, thanks to Lemma 1 we obtain a conjugated RDS given by ϕv(t,ω)v0:=T−1(θtω,ϕu(t,ω)T(ω,v0)) =T−1θtω,ϕu(t,ω)(S(0)−S0)eαz∗(ω) x(0)eαz∗(ω) =T−1(θtω,ϕu(t,ω)u0) =T−1(θtω,u(t;ω,u0)) =S0+σ(t)e−αz∗(θtω) κ(t)e−αz∗(θtω) =v(t;ω,v0) which means that ϕvis an RDS for our original stochastic system (5)-(6). Moreover, the global random attractor of the random system (9)-(10) A={A(ω)}ω∈Ω={(0,0)} becomes f A={e A(ω)}ω∈Ω={(S0,0)}, the global random attractor of the stochastic system (5)-(6). Stochastic chemostats 17 3.5 Numerical simulations and final comments To confirm the results above, in this section we show some numerical simulations for (3)-(4). We use the Euler-Maruyama method [15] considering an initial value (S0,x0) = (5,10),S0=1, D=3, a=0.6, m=3 and the following numerical scheme: Sj=Sj−1+f(xj−1,Sj−1)∆t+g(xj−1,Sj−1)·(W(τj)−W(τj−1)), xj=xj−1+e f(xj−1,Sj−1)∆t+e g(xj−1,Sj−1)·(W(τj)−W(τj−1)), where we define functions f,g,e fand e gas f(xj−1,Sj−1) = (S0−Sj−1)D−mSj−1xj−1 a+Sj−1, g(xj−1,Sj−1) = α(S0−Sj−1), e f(xj−1,Sj−1) = xj−1mSj−1 a+Sj−1 −D, e g(xj−1,Sj−1) = αxj−1, and W(τj)−W(τj−1) = jR ∑ k=jR−R+1 dWk, where Ris a nonnegative integer number and dWkare N(0,1)−distributed independent random variables which can be generated numerically by pseudorandom number generators. From now on, the red lines in the pictures represent the stochastic solutions of system (3)-(4) and the blue ones the deterministic solutions of the same system. By the previous sections, we know that system (3)-(4) possesses a random attractor given by ˜ A={(S0,0)}as long as (11) is satisfied. For the following different values of αwe obtain the following values of ˜ λ: (a) Case α=0.1: eλ:=e Da m−e D=359.4≥1=S0. 18 T. Caraballo, M.J. Garrido-Atienza, J. L´ opez-de-la-Cruz (b) Case α=0.5: eλ:=e Da m−e D=13.8≥1=S0. (c) Case α=1: eλ:=e Da m−e D=3≥1=S0. (d) Case α=1.5: eλ:=e Da m−e D=1≥1=S0. Summing up, in all the above cases eλ≥S0and D≥α2 2hold, hence the solutions of system (3)-(4) for the previous values of the parameters go to (S0,0)=(1,0), the random attractor. The following pictures show what we expected from the theory and numerical computing and we also can observe what happens when the intensity of noise increases. 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 S(t) 0123456 x(t) 0 2 4 6 8 10 12 Phase plane Fig. 1 α=0.1 on the left and α=0.5 on the right S(t) -10123456 x(t) 0 2 4 6 8 10 12 14 16 Phase plane S(t) -10123456 x(t) 0 2 4 6 8 10 12 Phase plane Fig. 2 α=1 on the left and α=1.5 on the right Stochastic chemostats 19 However, the next pictures show what happens if eλ<S0holds true. In this case D=1.5 instead of D=3 as in the previous cases. Fig. 3 α=0.1 on the left and α=0.5 on the right Fig. 4 α=0.7 on the left and α=0.9 on the right Remark 3. We would like to mention that the fact that the substrate S(or its corresponding σ) may take negative values does not produce any mathematical inconsistence in our analysis, in other words, our mathematical analysis is accurate to handle the mathematical problem. However, from a biological point of view, this may reflect some troubles and suggests that either the fact of perturbing the dilution rate with an additive noise may not be a realistic situation, or that we should try to use a some kind of switching system to model our real chemostat in such a way that when the dilution may be negative we use a different equation to model the system. This will lead us to a different analysis in some subsequent papers by considering a different kind of randomness or stochasticity in this parameter or designing a different model for our problem. 20 T. Caraballo, M.J. Garrido-Atienza, J. L´ opez-de-la-Cruz On the other hand, it could also be considered a noisy term in each equation of the deterministic model in the same fashion as in the paper by Imhof and Walcher [16], which ensures the positivity of both the nutrient and biomass, although does not preserve the wash out equilibrium from the deterministic to the stochastic model. 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