# Asymptotic Behavior of a Chemostat Model with Stochastic Perturbation on the Dilution Rate

Authors: Chaoqun Xu; Sanling Yuan; Tonghua Zhang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We present a stochastic
simple chemostat model in which the dilution rate was influenced by
white noise. The long time behavior of the system is studied.
Mainly, we show how the solution spirals around the washout
equilibrium and the positive equilibrium of deterministic system
under different conditions. Furthermore, the sufficient conditions
for persistence in the mean of the stochastic system and washout of
the microorganism are obtained. Numerical simulations are carried
out to support our results.


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## Body

## 1. Introduction

Modeling microbial growth is a problem of special interest in mathematical biology and theoretical ecology. One particular class of models includes deterministic models of microbial growth in the chemostat. Equations of the basic chemostat model with a single species and a single substrate take the form(1)dSdt=(S0-S)D-xγp(S),dxdt=x(p(S)-D),
where S(t) and x(t), respectively, denote concentrations of the nutrient and the microbial biomass, and all the parameters are positive constants; S0 denotes the feed concentration of the nutrient and D denotes the volumetric dilution rate (flow rate/volume). The function p(S) denotes the microbial growth rate and a typical choice for p(S) is p(S)=(mS)/(a+S) [1]. The stoichiometric yield coefficient γ denotes the ratio of microbial biomass produced to the mass of the nutrient consumed.The dynamics of the basic model is simple. Ifγ is constant and p(S) is a monotonically increasing function, then the microorganism can either become extinct or persist at an equilibrium level [2–5]. The particular outcome depends solely on the break-even concentration λ where p(λ)=D. Specifically, if λ<S0, the organism persists, and if λ≥S0, it becomes extinct.Note that the modeling process that leads to (1) relies on the fact that the stochastic effects can be neglected or averaged out, thanks to the law of large numbers. This is possible only at macroscopic scale, for large population sizes, and under homogeneity conditions. At all other scales or when the homogeneity conditions are not met, random effects cannot be neglected, just as that stated in Campillo et al. [6]: “This is the case at microscopic scale, in small population size, as well as all scales preceding the one where (1) is valid. This is also when the homogeneity condition is not met, for example, in unstirred conditions. Also the accumulation of small perturbations in the context of multispecies could not be neglected. Moreover, whereas the experimental results observed in well mastered laboratory conditions match closely the ODE theoretical behavior, a noticeable difference may occur in operational.” So, even if the description (1) is sufficient for a number of applications of interest, it does not account for the stochastic aspects of the problem (see Campillo et al. [6] for more details on this respect).In fact, ecosystem dynamics is inevitably affected by environmental white noise which is an important component in realism. It is therefore useful to reveal how the noise affects the ecosystem dynamics. As a matter of fact, stochastic biological systems and stochastic epidemic models have recently been studied by many authors; see [7–16]. We also refer the readers to Imhof and Walcher [7] for the reason why the stochastic effects should be considered in chemostat modeling.Taking into account the effect of randomly fluctuating environment, we introduce randomness into model (1) by replacing the dilution rate D by D→D+αB˙(t), where B˙(t) is a white noise (i.e., B(t) is a Brownian motion) and α≥0 represents the intensity of noise. This is only a first step in introducing stochasticity into the model. Ideally we would also like to introduce stochastic environmental variation into the other parameters such as the feed nutrient concentration S0 and the microbial growth rate p(S), but to do this would make the analysis too difficult. As a result, model (1) becomes the following system of stochastic differential equation:
(2)dS=[(S0-S)D-xγp(S)]dt+α(S0-S)dB(t),dx=x(p(S)-D)dt-αxdB(t).To the best of our knowledge, a very little amount of work has been done with the above model. Motivated by this, in this paper, we will investigate the long time behavior of model (2) with p(S) taking the Holling II functional response function, that is, p(S)=(mS)/(a+S).We should point out that a few papers have already addressed the stochastic modeling of the chemostat [6, 7, 17–19]. Here we only mention a recent paper by Imhof and Walcher [7]. They introduced a variant of the deterministic single-substrate chemostat model for which the persistence of all species is possible. To derive a stochastic model they considered a discrete-time Markov process with jumps corresponding to the deterministic system added with a centered Gaussian term, letting the time step converges to zero leads to a system of stochastic differential equations. They proved that random effects may lead to extinction in scenarios where the deterministic model predicts persistence; they also established some stochastic persistence results. Obviously, the model they considered is different from our model (2). We also refer the readers to Campillo et al. [6] for other works on the stochastic modeling of the chemostat.The organization of this paper is as follows. The model and some basic results about the model are presented in the next section. In Section3, the long time behavior of the stochastic model, including the asymptotic behavior around the positive equilibrium point and that around the extinction equilibrium point, is analyzed. An extinction result on the model is presented in Section 4. Finally, in Section 5, simulations and discussions are presented.


## 2. The Model and Some Fundamental Results

Assuming in model (2) that p(S) takes the Holling II functional response function, that is, p(S)=(mS)/(a+S), consider the following stochastic chemostat model:
(3)dS=[(S0-S)D-mSxa+S]dt+α(S0-S)dB(t),dx=x(mSa+S-D)dt-αxdB(t),
where S(t) and x(t) have the similar biological meanings as in model (1), and all parameters are positive constants. S0, D, and α play similar roles as in model (2), and m and a are the maximal growth rates of the organism and the Michaelis-Menten (or half saturation) constant, respectively; the yield coefficient γ has been scaled out by scaling.AsS(t) and x(t) in (3) are concentrations of the substrate and the microorganism at time t, respectively, we are only interested in the positive solutions. Moreover, in order for a stochastic differential equation to have a unique global (i.e., no explosion in a finite time) solution for any given initial value, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition. However, the coefficients of (3) do not satisfy the linear growth condition, though they are locally Lipschitz continuous, so the solution of model (3) may explode at a finite time [20, 21]. In the following, by using the comparison theorem for stochastic equation (see [8, 22]), we first show the solution of model (3) is positive and global.Theorem 1.
For any given initial value(S0,x0)∈ℝ+2, there is a unique positive solution (S(t),x(t)) to model (3) on t≥0, and the solution will remain in ℝ+2 with probability one, namely, (S(t),x(t))∈ℝ+2 for all t≥0 almost surely.Proof.
Since the coefficients of model (3) are locally Lipschitz continuous for any given initial value (S0,x0)∈ℝ+2, there is a unique positive local solution (S(t),x(t)) on t∈[0,τe), where τe is the explosion time (i.e., the moment at which the solution tends to infinity). Next, we show that τe=∞.
Since the solution is positive, we have(4)dS≤(S0-S)(Ddt+αdB(t)),dx≤x(m-D)dt-αxdB(t).
Denote byΦ(t) the solution of the following stochastic differential equation:
(5)dΦ=(S0-Φ)(Ddt+αdB(t)),Φ(0)=S0,
and Ψ(t) the solution of the equation
(6)dΨ=(m-D)Ψdt-αΨdB(t),Ψ(0)=x0.
By the comparison theorem for stochastic equations, we haveS(t)≤Φ(t), x(t)≤Ψ(t), and t∈[0,τe), a.s.
Similarly, we can get(7)dS≥[(S0-S)D-mΨ]dt+α(S0-S)dB(t),dx≥-x(Ddt+αdB(t)).
Denote byφ the solution of stochastic differential equation
(8)dφ=[(S0-φ)D-mΨ]dt+α(S0-φ)dB(t),φ(0)=S0,
andψ the solution of the equation
(9)dψ=-ψ(Ddt+αdB(t)),ψ(0)=x0.
We have thatS(t)≥φ(t), x(t)≥ψ(t), t∈[0,τe), a.s.
To sum up, we have that(10)φ(t)≤S(t)≤Φ(t),ψ(t)≤x(t)≤Ψ(t),t∈[0,τe),a.s.
Noting that (5)–(9) are all linear stochastic differential equations, Φ(t), Ψ(t), φ(t), and ψ(t) can be explicitly solved from them, separately. Obviously, they are all positive and globally existent for all t∈[0,∞). From (10), we can have that τe=∞. The proof is thus completed.The following theorem shows that the solution(S(t),x(t)) of model (3) with any positive initial value is uniformly bounded in mean.Theorem 2.
For any given initial value(S0,x0)∈ℝ+2, the solution (S(t),x(t)) of model (3) has the property:
(11)limt→∞E(S(t)+x(t))=S0.Proof.
Define the functionV(t)=S(t)+x(t), by the Itô formula, we get
(12)dV=(S0-V(S,x))(Ddt+αdB).
Integrating both sides from 0 tot, and then taking expectations, yields
(13)EV(t)=V(0)+∫0t‍(S0-EV(s))Dds.
Consequently,(14)dEV(t)dt=(S0-EV(t))D.
It is clear that(15)limt→∞EV(t)=S0.
Thus, we complete the proof of Theorem2.


## 3. The Long Time Behavior of Model(3)

Whenα=0, model (3) becomes its corresponding deterministic system
(16)dS=[(S0-S)D-mSxa+S]dt,dx=x(mSa+S-D)dt.We have known that the dynamic behavior of model (16) is completely determined by the break-even concentration λ=Da/(m-D). Model (16) always has a washout equilibrium E0=(S0,0). When λ≥S0, E0 is globally asymptotically stable, and when 0<λ<S0, E0 loses its stability and a globally asymptotically stable positive equilibrium E*=(S*,x*) appears, where S*=λ and x*=S0-λ.Whenα≠0, E0 is still an equilibrium of model (3) but E* is not; that is to say, model (3) has no positive equilibrium. It is natural to ask whether the microorganism will persist or go to extinction in the chemostat. In this section we mainly use the way of estimating the oscillation around E* (or E0) to reflect how the solution of model (3) spirals around E* (or E0). In the following part of this section, we always denote δ0 by
(17)δ0=D-α22.


### 3.1. Asymptotic Behavior around the Washout EquilibriumE0

As mentioned above, system (16) always has an equilibrium E0 which is globally asymptotically stable provided λ≥S0. It is natural to ask whether the solutions of system (3) will be close to E0. We have the following theorem.Theorem 3.
Ifλ≥S0 and δ0>0, then the washout equilibrium E0 of system (3) is stochastically asymptotically stable in the large.Proof.
Define a functionV:ℝ+2→ℝ+ by
(18)V(S,x)=(S+x-S0)2+4δ0amx.
Obviously, the functionV is positive definite, and
(19)dV=LVdt-α[2(S+x-S0)2+4δ0amx]dB,
where,
(20)LV=-2D(S+x-S0)2+α2(S0-S)2-2α2(S0-S)x+α2x2+4δ0am(mSa+S-D)x=-2δ0(S+x-S0)2+4δ0am(mSa+S-D)x=-2δ0[(S-S0)2+x2]+4δ0am(mSa+S-D)x-4δ0(S-S0)x.
We will show that for all solutions(S(t),x(t)) of system (3) with initial value (S0,x0)∈ℝ+2,
(21)LV≤-2δ0a2(a+S0)2[(S-S0)2+x2].
Suppose first thatλ≥S0, then
(22)mSa+S-D≤mSa+S-mS0a+S0≤ma(S-S0).
Obviously,(23)4δ0am(mSa+S-D)x-4δ0(S-S0)x≤0.
It then follows from (20) that
(24)LV≤-2δ0[(S-S0)2+x2].
Suppose next thatS≤S0; noting also that λ≥S0, we have
(25)mSa+S-D≤mSa+S-mS0a+S0≤ma(a+S0)2(S0-S).
Therefore,(26)4δ0am(mSa+S-D)x-4δ0(S-S0)x≤4δ0[1-a2(a+S0)2](S0-S)x≤2δ0[1-a2(a+S0)2][(S-S0)2+x2].
It follows from (20) that
(27)LV≤-2δ0a2(a+S0)2[(S-S0)2+x2].
To sum up, (21) holds. That is to say, LV is negative definite. By Theorem A in the appendix, the washout equilibrium E0 of system (3) is stochastically asymptotically stable in the large. The proof of Theorem 3 is thus completed.


### 3.2. Asymptotic Behavior around the Positive EquilibriumE*

Theorem 4.
If0<λ<S0 and δ0>0, then for any solution (S(t),x(t)) of system (3) with initial value (S0,x0)∈ℝ+2,
(28)limsupt→∞1t∫0t‍[(S(s)-S*)2+(x(s)-x*)2]ds≤2(a+S*+4x*)2x*maα2.Proof.
The proof is inspired by the method of Imhof and Walcher [7]. Define a function V:ℝ+2→ℝ+ by
(29)V(S,x)=(S+x-S0)2+4δ0am(x-x*-x*lnxx*).
Obviously, the functionV is positive definite. Using Itô's formula we get
(30)dV=LVdt-2α[(S+x-S0)2+2δ0am(x-x*)]dB,
where
(31)LV=-2δ0(S+x-S0)2+4δ0am(mSa+S-D)(x-x*)+2δ0amα2x*=-2δ0[(S-S*)2+(x-x*)2]+2δ0amα2x*+4δ0am(mSa+S-D)(x-x*)-4δ0(S-S*)(x-x*).
We will show that for all solutions(S(t),x(t)) with initial value (S0,x0)∈ℝ+2,
(32)LV≤-δ0a2(a+S*+4x*)2[(S-S*)2+(x-x*)2]+2δ0amα2x*.
Suppose first that(S-S*)(x-x*)≥0, then (mS/(a+S)-D)(x-x*)≥0 and
(33)(mSa+S-D)(x-x*)=|mSa+S-D||x-x*|≤ma|S-S*||x-x*|=ma(S-S*)(x-x*).
Obviously,(34)4δ0am(mSa+S-D)(x-x*)-4δ0(S-S*)(x-x*)≤0.
By (31) we have
(35)LV≤-2δ0[(S-S*)2+(x-x*)2]+2δ0amα2x*.
Suppose next that(S-S*)(x-x*)<0 and S>S*+4x* (which implies that S-S*>0, x-x*<0), then
(36)-4δ0(S-S*)(x-x*)=4δ0(S-S*)(x*-x)≤4δ0(S-S*)x*≤δ0(S-S*)2.
By (31), we have
(37)LV≤-δ0(S-S*)2-2δ0(x-x*)2+2δ0amα2x*≤-δ0[(S-S*)2+(x-x*)2]+2δ0amα2x*.
Finally, if(S-S*)(x-x*)<0 and S≤S*+4x*, then
(38)4δ0am(mSa+S-D)(x-x*)-4δ0(S-S*)(x-x*)=4δ0|S-S*||x-x*|-4δ0am|mSa+S-D||x-x*|≤4δ0|S-S*||x-x*|-4δ0a2(a+S*+4x*)2|S-S*||x-x*|=4δ0[1-a2(a+S*+4x*)2]|S-S*||x-x*|≤2δ0[1-a2(a+S*+4x*)2][(S-S*)2+(x-x*)2].
By (31) we have
(39)LV≤-2δ0a2(a+S*+4x*)2[(S-S*)2+(x-x*)2]+2δ0amα2x*.
To sum up, (32) holds. Therefore,
(40)dV≤{-δ0a2(a+S*+4x*)2[(S-S*)2+(x-x*)2]5+2δ0amα2x*}dt5-2α[(S+x-S0)2+2δ0am(x-x*)]dB.
Integrating both sides from0 to t  yields
(41)V(t)-V(0)≤∫0t{-δ0a2(a+S*+4x*)2×[(S(s)-S*)2+(x(s)-x*)2]+2δ0amα2x*}ds+M(t),
where M(t)=-2α∫0t‍[(S(s)+x-S0)2+(2δ0a/m)(x(s)-x*)]dB. Obviously, M(t) is a local continuous martingale. From (3), we can get
(42)d(S+x-S0)=(S0-S-x)Ddt+(S0-S-x)αdB=(S+x-S0)(-Ddt-αdB).
It follows that
(43)S+x-S0=(S0+x0-S0)exp{(-D-α22)t-αB(t)}≤Cexp{-αB(t)},
where C=|S0+x0-S0|. Thus we have that
(44)〈M(t),M(t)〉t=4α2∫0t‍[(S(s)+x(s)-S0)2+2δ0am(x(s)-x*)]2ds≤4α2∫0t‍[C2exp{-2αmin0≤s≤tB(s)}+2δ0am×(x*+S0+Cexp{-αmin0≤s≤tB(s)})]2ds,
which implies that
(45)limsupt→∞〈M(t),M(t)〉tt<∞.
By Strong Law of Large Numbers (see Mao [21]), we obtain
(46)limt→∞M(t)t=0a.s.
It follows from (41) that
(47)limsupt→∞1t∫0t‍{-δ0a2(a+S*+4x*)2×[(S(s)-S*)2+(x(s)-x*)2]+2δ0amα2x*}ds≥0,
namely,
(48)limsupt→∞1t∫0t‍[(S(s)-S*)2+(x(s)-x*)2]ds≤2(a+S*+4x*)2x*amα2.
This completes the proof of Theorem4.Remark 5.
Theorems3 and 4 show that the solutions of system (3) will fluctuate, respectively, around E0 and E* when α is small enough such that δ0>0 (i.e., α<2D).


### 3.3. Persistence of System (3)

From the result of Theorem4, we conclude that system (3) is persistent when α is small, which implies the persistence of microorganism. L. S. Chen and J. Chen in [23] proposed the definition of persistence in the mean for the deterministic system. Here, we also use this definition for the stochastic system (see also Ji et al. [24]).Definition 6.
System (3) is said to be persistent in the mean, if
(49)liminft→∞1t∫0t‍S(s)ds>0,liminft→∞1t∫0t‍x(s)ds>0a.s.Theorem 7.
Let(S(t),x(t)) be the solution of system (3) with initial value (S0,x0)∈ℝ+2. If 0<λ<S0, δ0>0, and
(50)α2<min{am(S*)22(a+S*+4x*)2x*,amx*2(a+S*+4x*)2},
then system (3) is persistent in the mean.Proof.
Obviously, (28) holds. It follows that
(51)lim supt→∞1t∫0t‍(S(s)-S*)2ds≤2(a+S*+4x*)2x*amα2a.s.,(52)limsupt→∞1t∫0t‍(x(s)-x*)2ds≤2(a+S*+4x*)2x*amα2a.s.
Noting also that(53)2(S*)2-2S*S=2S*(S*-S)≤(S*)2+(S-S*)2,
we have
(54)S≥S*2-(S-S*)22S*.
It follows from (54), (50), and (51) that
(55)liminft→∞1t∫0t‍S(s)dslim≥S*2-limsupt→∞1t∫0t‍(S(s)-S*)22S*dslim≥S*2-2(a+S*+4x*)2x*2amS*α2>0a.s.
Similarly, we have(56)liminft→∞1t∫0t‍x(s)dslim≥x*2-limsupt→∞1t∫0t‍(x(s)-x*)22x*dslim≥x*2-2(a+S*+4x*)2x*2amx*α2>0a.s.
Therefore, system (3) is persistent in the mean.


## 3.1. Asymptotic Behavior around the Washout EquilibriumE0

As mentioned above, system (16) always has an equilibrium E0 which is globally asymptotically stable provided λ≥S0. It is natural to ask whether the solutions of system (3) will be close to E0. We have the following theorem.Theorem 3.
Ifλ≥S0 and δ0>0, then the washout equilibrium E0 of system (3) is stochastically asymptotically stable in the large.Proof.
Define a functionV:ℝ+2→ℝ+ by
(18)V(S,x)=(S+x-S0)2+4δ0amx.
Obviously, the functionV is positive definite, and
(19)dV=LVdt-α[2(S+x-S0)2+4δ0amx]dB,
where,
(20)LV=-2D(S+x-S0)2+α2(S0-S)2-2α2(S0-S)x+α2x2+4δ0am(mSa+S-D)x=-2δ0(S+x-S0)2+4δ0am(mSa+S-D)x=-2δ0[(S-S0)2+x2]+4δ0am(mSa+S-D)x-4δ0(S-S0)x.
We will show that for all solutions(S(t),x(t)) of system (3) with initial value (S0,x0)∈ℝ+2,
(21)LV≤-2δ0a2(a+S0)2[(S-S0)2+x2].
Suppose first thatλ≥S0, then
(22)mSa+S-D≤mSa+S-mS0a+S0≤ma(S-S0).
Obviously,(23)4δ0am(mSa+S-D)x-4δ0(S-S0)x≤0.
It then follows from (20) that
(24)LV≤-2δ0[(S-S0)2+x2].
Suppose next thatS≤S0; noting also that λ≥S0, we have
(25)mSa+S-D≤mSa+S-mS0a+S0≤ma(a+S0)2(S0-S).
Therefore,(26)4δ0am(mSa+S-D)x-4δ0(S-S0)x≤4δ0[1-a2(a+S0)2](S0-S)x≤2δ0[1-a2(a+S0)2][(S-S0)2+x2].
It follows from (20) that
(27)LV≤-2δ0a2(a+S0)2[(S-S0)2+x2].
To sum up, (21) holds. That is to say, LV is negative definite. By Theorem A in the appendix, the washout equilibrium E0 of system (3) is stochastically asymptotically stable in the large. The proof of Theorem 3 is thus completed.


## 3.2. Asymptotic Behavior around the Positive EquilibriumE*

Theorem 4.
If0<λ<S0 and δ0>0, then for any solution (S(t),x(t)) of system (3) with initial value (S0,x0)∈ℝ+2,
(28)limsupt→∞1t∫0t‍[(S(s)-S*)2+(x(s)-x*)2]ds≤2(a+S*+4x*)2x*maα2.Proof.
The proof is inspired by the method of Imhof and Walcher [7]. Define a function V:ℝ+2→ℝ+ by
(29)V(S,x)=(S+x-S0)2+4δ0am(x-x*-x*lnxx*).
Obviously, the functionV is positive definite. Using Itô's formula we get
(30)dV=LVdt-2α[(S+x-S0)2+2δ0am(x-x*)]dB,
where
(31)LV=-2δ0(S+x-S0)2+4δ0am(mSa+S-D)(x-x*)+2δ0amα2x*=-2δ0[(S-S*)2+(x-x*)2]+2δ0amα2x*+4δ0am(mSa+S-D)(x-x*)-4δ0(S-S*)(x-x*).
We will show that for all solutions(S(t),x(t)) with initial value (S0,x0)∈ℝ+2,
(32)LV≤-δ0a2(a+S*+4x*)2[(S-S*)2+(x-x*)2]+2δ0amα2x*.
Suppose first that(S-S*)(x-x*)≥0, then (mS/(a+S)-D)(x-x*)≥0 and
(33)(mSa+S-D)(x-x*)=|mSa+S-D||x-x*|≤ma|S-S*||x-x*|=ma(S-S*)(x-x*).
Obviously,(34)4δ0am(mSa+S-D)(x-x*)-4δ0(S-S*)(x-x*)≤0.
By (31) we have
(35)LV≤-2δ0[(S-S*)2+(x-x*)2]+2δ0amα2x*.
Suppose next that(S-S*)(x-x*)<0 and S>S*+4x* (which implies that S-S*>0, x-x*<0), then
(36)-4δ0(S-S*)(x-x*)=4δ0(S-S*)(x*-x)≤4δ0(S-S*)x*≤δ0(S-S*)2.
By (31), we have
(37)LV≤-δ0(S-S*)2-2δ0(x-x*)2+2δ0amα2x*≤-δ0[(S-S*)2+(x-x*)2]+2δ0amα2x*.
Finally, if(S-S*)(x-x*)<0 and S≤S*+4x*, then
(38)4δ0am(mSa+S-D)(x-x*)-4δ0(S-S*)(x-x*)=4δ0|S-S*||x-x*|-4δ0am|mSa+S-D||x-x*|≤4δ0|S-S*||x-x*|-4δ0a2(a+S*+4x*)2|S-S*||x-x*|=4δ0[1-a2(a+S*+4x*)2]|S-S*||x-x*|≤2δ0[1-a2(a+S*+4x*)2][(S-S*)2+(x-x*)2].
By (31) we have
(39)LV≤-2δ0a2(a+S*+4x*)2[(S-S*)2+(x-x*)2]+2δ0amα2x*.
To sum up, (32) holds. Therefore,
(40)dV≤{-δ0a2(a+S*+4x*)2[(S-S*)2+(x-x*)2]5+2δ0amα2x*}dt5-2α[(S+x-S0)2+2δ0am(x-x*)]dB.
Integrating both sides from0 to t  yields
(41)V(t)-V(0)≤∫0t{-δ0a2(a+S*+4x*)2×[(S(s)-S*)2+(x(s)-x*)2]+2δ0amα2x*}ds+M(t),
where M(t)=-2α∫0t‍[(S(s)+x-S0)2+(2δ0a/m)(x(s)-x*)]dB. Obviously, M(t) is a local continuous martingale. From (3), we can get
(42)d(S+x-S0)=(S0-S-x)Ddt+(S0-S-x)αdB=(S+x-S0)(-Ddt-αdB).
It follows that
(43)S+x-S0=(S0+x0-S0)exp{(-D-α22)t-αB(t)}≤Cexp{-αB(t)},
where C=|S0+x0-S0|. Thus we have that
(44)〈M(t),M(t)〉t=4α2∫0t‍[(S(s)+x(s)-S0)2+2δ0am(x(s)-x*)]2ds≤4α2∫0t‍[C2exp{-2αmin0≤s≤tB(s)}+2δ0am×(x*+S0+Cexp{-αmin0≤s≤tB(s)})]2ds,
which implies that
(45)limsupt→∞〈M(t),M(t)〉tt<∞.
By Strong Law of Large Numbers (see Mao [21]), we obtain
(46)limt→∞M(t)t=0a.s.
It follows from (41) that
(47)limsupt→∞1t∫0t‍{-δ0a2(a+S*+4x*)2×[(S(s)-S*)2+(x(s)-x*)2]+2δ0amα2x*}ds≥0,
namely,
(48)limsupt→∞1t∫0t‍[(S(s)-S*)2+(x(s)-x*)2]ds≤2(a+S*+4x*)2x*amα2.
This completes the proof of Theorem4.Remark 5.
Theorems3 and 4 show that the solutions of system (3) will fluctuate, respectively, around E0 and E* when α is small enough such that δ0>0 (i.e., α<2D).


## 3.3. Persistence of System (3)

From the result of Theorem4, we conclude that system (3) is persistent when α is small, which implies the persistence of microorganism. L. S. Chen and J. Chen in [23] proposed the definition of persistence in the mean for the deterministic system. Here, we also use this definition for the stochastic system (see also Ji et al. [24]).Definition 6.
System (3) is said to be persistent in the mean, if
(49)liminft→∞1t∫0t‍S(s)ds>0,liminft→∞1t∫0t‍x(s)ds>0a.s.Theorem 7.
Let(S(t),x(t)) be the solution of system (3) with initial value (S0,x0)∈ℝ+2. If 0<λ<S0, δ0>0, and
(50)α2<min{am(S*)22(a+S*+4x*)2x*,amx*2(a+S*+4x*)2},
then system (3) is persistent in the mean.Proof.
Obviously, (28) holds. It follows that
(51)lim supt→∞1t∫0t‍(S(s)-S*)2ds≤2(a+S*+4x*)2x*amα2a.s.,(52)limsupt→∞1t∫0t‍(x(s)-x*)2ds≤2(a+S*+4x*)2x*amα2a.s.
Noting also that(53)2(S*)2-2S*S=2S*(S*-S)≤(S*)2+(S-S*)2,
we have
(54)S≥S*2-(S-S*)22S*.
It follows from (54), (50), and (51) that
(55)liminft→∞1t∫0t‍S(s)dslim≥S*2-limsupt→∞1t∫0t‍(S(s)-S*)22S*dslim≥S*2-2(a+S*+4x*)2x*2amS*α2>0a.s.
Similarly, we have(56)liminft→∞1t∫0t‍x(s)dslim≥x*2-limsupt→∞1t∫0t‍(x(s)-x*)22x*dslim≥x*2-2(a+S*+4x*)2x*2amx*α2>0a.s.
Therefore, system (3) is persistent in the mean.


## 4. Washout of the Organism in the Chemostat

The following theorem shows that sufficiently large noise can make the microorganism extinct exponentially with probability one in a simple chemostat.Theorem 8.
For any given initial value(S0,x0)∈ℝ+2, the solution (S(t),x(t)) of system (3) has the property:
(57)limsupt→∞lnx(t)t≤(m-D)-12α2.Proof.
Define the functionV(x)=lnx; by the Itô formula, we get
(58)dV=1xdx-12x2(dx)2=(mSa+S-D)dt-αdB-12α2dt≤(m-D-12α2)dt-αdB.
Integrating both sides from 0 tot yields
(59)lnx(t)-lnx(0)≤(m-D-12α2)t-αB(t).
Dividingt on the both sides and letting t→∞, we have
(60)limsupt→∞lnx(t)t≤(m-D)-12α2.
The proof of Theorem8 is completed.Remark 9.
In fact, we can see from the proof of Theorem8 that only the boundedness of p(S)=mS/(a+S) is used. So, Theorem 8 holds true for a much larger class bounded growth rate functions, for example, p(S)=mS/(a+bS+S2) (see also Figure 4).


## 5. Simulations and Discussions

In order to confirm the results above, we numerically simulate the solution of system (3) and system (16) with the initial (S0,x0)=(0.7,0.3). The numerical simulation is given by the following Milstein scheme [25]. Consider the discretization of system (3) for t=0,Δt,2Δt,…,nΔt:
(61)Si+1=Si+[(S0-Si)D-mSixia+Si]Δt+α(S0-Si)Δtξi,xi+1=xi+xi(mSia+Si-D)Δt-αxiΔtξi,
where time increment Δt>0, and ξi is N(0,1)-distributed independent random variables which can be generated numerically by pseudorandom number generators. In Figures 1–4, we will use the blue lines and the red lines to represent the solutions of deterministic system (16) and those of stochastic system (3), respectively.Deterministic and stochastic trajectories of simple chemostat model for initial conditionS0=0.7; x0=0.3; and S0=1, m=2, a=0.6, and D=1.3.(a) α=0.2, (b) α=0.5.
(a)
(b)Deterministic and stochastic trajectories of simple chemostat model for initial conditionS0=0.7; x0=0.3; and S0=1, m=2, a=0.6, and D=0.8.(a) α=0.1, (b) α=0.02.
(a)
(b)Figure 3
Deterministic and stochastic trajectories of simple chemostat model for initial conditionS0=0.7; x0=0.3; and S0=1, m=2, a=0.6, D=0.8, and α=1.55.Figure 4
Deterministic and stochastic trajectories of chemostat model with Holling IV functional response function for initial conditionS0=0.7; x0=0.3; and S0=1, m=2, a=0.6, b=0.4, D=0.8, and α=1.55.By Theorem3, we expect that the washout equilibrium E0 of system (3) is globally asymptotically stable under the conditions λ≥S0 and δ0>0. In Figure 1, we choose parameters S0=1,m=2, a=0.6, D=1.3, and α=0.2 in Figure 1(a). We can compute that λ=39/35>1=S0 and δ0=1.28>0. Then the washout equilibrium E0 of systems (3) and (16) is globally asymptotically stable (see Figure 1(a)). Furthermore, to watch the influence of the intensity of noise on the dynamics of system (3), we take α=0.5 in Figure 1(b). We can see that the solutions of system (3) with α=0.5 tend to the washout equilibrium faster than that of system (3) with α=0.2.Next, we consider the effect of stochastic fluctuations of environment on the positive equilibrium of the corresponding deterministic system. As mentioned in the Section2, there is a positive equilibrium E* of system (16) when 0<λ<S0, and it is globally asymptotically stable. Besides, Theorem 4 tells us that the difference between the perturbed solution and E* is related to white noise under the conditions 0<λ<S0 and δ0>0. In systems (3) and (16), we choose parameters S0=1, m=2, a=0.6, and D=0.8. As for the washout equilibrium E0, we take α=0.1 in Figure 2(a) and α=0.02 in Figure 2(b). We can compute that λ=0.4<1=S0 and δ0=0.795>0 in Figure 2(a) and δ0=0.7998>0 in Figure 2(b). As expected, the solutions of system (3) are oscillating around the positive equilibrium E* for a long time (see Figure 2). Besides, we can observe that with white noise getting weaker, the fluctuation around E* gets smaller, which supports the result of Theorem 4.Theorem8 shows that the microorganism will be washed out eventually under the condition α2>2(m-D) even if system (16) has a positive equilibrium. As an example, we choose parameters S0=1, m=2, a=0.6, and D=0.8 in systems (3) and (16). We can compute that λ=0.4<1=S0; thus system (16) will persist. But when α is large enough, for example, we take α=1.55, we can compute that α2=2.4025>2.4=2(m-D), and the microorganism will die out (see Figure 3). This shows that strong noise may lead to the extinction of the species in the chemostat.To further confirm that Theorem8 holds for a much lager class of growth rate function (i.e., Remark 9), we take p(S) as the Holling IV functional response function p(S)=(mS)/(a+bS+S2) instead of p(S)=mS/(a+S) and all parameters have the similar values as above except for b=0.4 in systems (3) and (16). Numerical simulations show that they have the similar dynamics as for p(S)=mS/(a+S) (see Figure 4).Some interesting questions deserve further investigation. One may propose more realistic but complex models, such as incorporating the colored noise into the system. Moreover, it is interesting to study other parameters perturbation.



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