DYNAMICS OF A STOCHASTIC CHEMOSTAT COMPETITION MODEL WITH PLASMID-BEARING AND PLASMID-FREE ORGANISMS

In this paper, we consider a chemostat model of competition between plasmid-bearing and plasmid-free organisms, perturbed by white noise. Firstly, we prove the existence and uniqueness of the global positive solution. Then by constructing suitable Lyapunov functions, we establish sufficient conditions for the existence of a unique ergodic stationary distribution. Furthermore, conditions for extinction of plasmid-bearing organisms are obtained. Theoretical analysis indicates that large noise intensity σ 2 is detrimental to the survival of plasmid-bearing organisms and is not conducive to the commercial production of genetically altered organisms. Finally, numerical simulations are presented to illustrate the results.


Introduction
The chemostat, a continuous culture device mainly used for various theoretical studies related to the growth rate of microorganisms, plays an important role in waste treatment and fermentation processes [20]. It has the advantage that the parameters are readily measurable, and thus the mathematics is tractable [3]. Many types of chemostat models have been investigated extensively in the literature (see [1,2,12,17,19,26,38] as well as there references). Especially, competitive chemostat models have been studied by many researchers (see e.g. [18,21,22,27,31]).
Genetically altered organisms are frequently used to produce desired products. It has been widely used in agriculture, medicine, environmental protection and other fields. The alteration is accomplished by the insertion of a recombinant DNA into the cell in the form of a plasmid [5]. The plasmid-free organism is unencumbered by the added metabolic load the plasmid imposes, and thus may be a better competitor than plasmid-bearing organism. Moreover, the plasmid can be lost in the reproduction, resulting in a plasmid-free organism [6]. Since commercial production can take place on a scale of many generations, it is possible for the plasmid-free organism to take over the culture. The study of chemostat models for the competition between plasmid-bearing and plasmid-free organisms has received considerable attention (see e.g. [8,23,24,29,[32][33][34][35] and the references therein). For a chemostat with plasmid-bearing, plasmid-free organisms and periodically pulsed substrate, Xiang and Song [29] showed there exists a asymptotically stable two microorganisms extinction periodic solution. They also established sufficient conditions for the extinction of plasmid-bearing organism and permanence of the other microorganism. Using standard techniques of bifurcation theory, Shi et al. [24] proved the existence of positive periodic solution for a chemostat model with plasmid-bearing, plasmid-free organisms competition and impulsive effect. In [23], Stephanopoulis and Lapidus proposed the following chemostat competition model with Monod response functions where S(t), x 1 (t) and x 2 (t) stand for the concentrations of nutrient, plasmid-bearing and plasmid-free organisms at time t, respectively. S 0 is the original input concentration of nutrient and D is the common dilution rate. γ represents the yield constant. µ 1 and µ 2 are the maximum growth rates of plasmid-bearing and plasmid-free organisms, respectively. K 1 and K 2 are the corresponding half-saturation constants. q is the probability that a plasmid is lost in reproduction. Hus et al. [5] studied this model and provided a global analysis of the asymptotic behavior. However, in reality, chemostat systems are inevitably subject to environmental noise. To reveal the effect of white noise on the continuous culture of microorganisms, some authors have investigated the dynamics of stochastic chemostat systems (see e.g. [9,25,28,30,36,37]). For example, for a classical chemostat model in the stochastic environment, Zhao and Yuan [36] derived sharp conditions for the existence of stationary distribution by using the property of Feller process and concluded that noises have negative effects on persistence of the microorganism. Sun et al. [25] considered a stochastic two-species Monod competition chemostat model, which is subject to environmental noise. They analyzed the asymptotic behavior of the solutions. Zhang and Jiang [37] discovered sufficient conditions which guarantee that the principle of competitive exclusion holds for a stochastic chemostat model with Holling type II functional response. Inspired by the relevant works, we assume the environmental noise is proportional to the variables and consider the following stochastic chemostat model with plasmid-bearing, plasmid-free organisms competition. For the sake of simplicity, we use S, x 1 and x 2 to denote S(t), x 1 (t) and x 2 (t), respectively.
where the same notations are used as in (1.1). W i (t), i = 1, 2, 3 are standard onedimensional independent Brownian motion defined on a complete probability space (Ω, F , {F t } t≥0 , P) with a filtration {F t } t≥0 satisfying the usual conditions (i.e. it is right continuous and F 0 contains all P−null sets). β 2 i > 0 represents the intensity of W i (t).
Stationary distribution can enrich the dynamical behavior of the stochastic chemostat system. It not only means random weak stability, but also provides a better description of persistence [14], which gives us a deeper understanding of how environmental noise affects the steady state for persistence. To the authors best knowledge, there are few studies on stationary distribution of the stochastic chemostat model with plasmid-bearing, plasmid-free organisms competition in the existing literature. In this paper, we attempt to do some work in this area. Our main effort is to construct suitable Lyapunov functions and find a bounded domain so that the diffusion operator is negative outside the domain.
The organization of the paper is as follows. In the next section, we analyze model (1.2) and give a lemma, which is necessary for later discussion. For the equivalent system (2.1) of model (1.2), we prove the existence and uniqueness of the global positive solution in Section 3. In Section 4, sufficient conditions for the existence of a unique ergodic stationary distribution are established. In Section 5, we obtain conditions for extinction of plasmid-bearing organisms. In Section 6, numerical simulations are carried out to support the theoretical results and we make a further discussion.

Model analysis and preliminaries
The variables in system (1.2) may be rescaled by S 0 . Let Then system (1.2) is transformed into the following equations (replacing τ with t) So we can find out the dynamical properties of system (1.2) by studying above model. The corresponding deterministic system to This model has three equilibria E 1 : (1, 0, 0), E 2 : (s, 0,ȳ), wheres andȳ satisfy m2s About the properties of these three equilibria, the reader can refer to Table 1 in Ref. [5].
Next, we present a lemma [11] which gives a criterion for the existence of a unique ergodic stationary distribution. Let X(t) be a homogeneous Markov process in R l (R l represents euclidean l-space) satisfying the stochastic equation

The diffusion matrix is
m (x).
Then the Markov process X(t) has a unique stationary distribution π(·). Let f (x) be a function integrable with respect to the measure π. For all x ∈ R l , the following formula holds For simplicity, we denote

Existence and uniqueness of positive solution
To research the dynamical behavior of a chemostat model, the first concern is whether the solution is positive and global. In this section, we shall show that system (2.1) has a unique global positive solution for any given initial value by making use of the Lyapunov function method as mentioned in [15].
Since the coefficients of model (2.1) satisfy the local Lipschitz condition, for any given initial value (s(0), where τ e denotes the explosion time.
Next we show this solution is global. we only need to prove τ e = ∞ a.s. Let k 0 ≥ 1 be sufficiently large such that s(0), x(0) and y(0) all lie within the interval where throughout this paper, we set inf ∅ = ∞ (as usual ∅ denotes the empty set). Obviously, τ k is increasing as k → ∞. Set τ ∞ = lim k→+∞ τ k , whence τ ∞ ≤ τ e a.s. If we can show that τ ∞ = ∞ a.s., then τ e = ∞ a.s. and (s(t), x(t), y(t)) ∈ R 3 + a.s. for all t ≥ 0. In other words, in order to complete the proof, we need to show τ ∞ = ∞ a.s. If this statement is false, then there is a pair of constants T > 0 and ϵ ∈ (0, 1) such that So there is an integer k 1 ≥ k 0 such that where Integrating and taking the expectation yield By Growrall inequality, we have By (3.2), one can see that where I Ω k represents the indicator function of Ω k . Here k → ∞ leads to the contradiction ∞>+∞, so we must have τ ∞ =∞ a.s. This completes the proof.

Existence of ergodic stationary distribution
In this section, for system (2.1), we establish sufficient conditions for the existence of a unique ergodic stationary distribution, which implies the plasmid-bearing and plasmid-free organisms can coexist in the chemostat. Defineλ

1) is given by
which is positive definite for any compact subset of R 3 + . Clearly, (A1) in Lemma 2.1 holds. Now we are in the position to validate the condition (A2) of Lemma 2.1. We need to show there is a non-negative C 2 -function V and a bounded domain D ε ⊂ R 3 + such that LV is negative for any (s, x, y) ∈ R 3 + \ D ε . Define a C 2 -function V 1 : R 3 + → R as follows By Itô's formula, the basic inequality (a+b) 2 ≤ 2(a 2 +b 2 ),s+ m2sȳ a2+s = 1 and m2s a2+s = 1, we get . From (4.1)-(4.3), it then follows that By virtue of Young inequality, we obtain This, together with (4.4), yields Define a C 2 -function V 2 (s) = − log s. Applying Itô's formula, we calculate that Define a C 2 -function V 3 (y) = − log y. Then Define a C 2 -function V 4 : R 3 + → R in the following form where θ is a constant satisfying 0 < θ < . Using Itô's formula leads to where Then we define a C 2 -function F : and It is easy to check that lim inf k→∞,(s,x,y)∈R 3 , k). Furthermore, F is a continuous function. Hence, F (s, x, y) has a minimum point F (s 0 , x 0 , y 0 ) in the interior of R 3 + .
According to the above analysis, we construct a non-negative C 2 -function V : By (4.5)-(4.8) and (4.10), we derive Define a bounded closed domain where ε > 0 is a sufficiently small constant. In the set R 3 + \ D ε , we can choose ε sufficiently small such that the following conditions hold where K 3 , K 4 are constants which can be found from (4.16) and (4.17). For convenience, we divide R 3 + \ D ε into six domains, Next we will show that LV (s, x, y) ≤ −1 on R 3 + \ D ε , which is equivalent to proving it on the above six domains, respectively. Case 1. If (s, x, y) ∈ D 1 , by (4.11) and (4.12), one can derive (4.16) Therefore, LV (s, x, y) ≤ −1 for any (s, x, y) ∈ D 1 .
Based on the above six situations, we can conclude that for a sufficiently small ε, The proof is complete.

Extinction
In this section, we investigate the extinction of plasmid-bearing organisms in system (2.1). First of all, we give two useful lemmas.
Proof. An application of Itô's formula yields Integrating this from 0 to t and dividing by t on both sides result in The strong law of large numbers [10] implies that In view of (5.1) and (5.3), we obtain Using Itô's formula, one can derive From (5.5) and (5.6), it follows that Thus, Then it is easy to see that Taking the inferior limit on both sides of (5.8) and combining with Lemma 5.1, from (5.3) and (5.4), we obtain lim inf This completes the proof.

Numerical simulations and discussion
This paper is devoted to a chemostat model with plasmid-bearing, plasmid-free organisms competition, which is disturbed by white noise. We first prove the system has a unique global positive solution for any initial value. Then, using the boundary equilibrium point E 2  2 , then the plasmid-bearing organism will go extinct exponentially with probability one and the plasmid-free organism will survive. This implies large noise intensity σ 2 2 is detrimental to the survival of plasmid-bearing organisms. In this case, the plasmid-free organism will take over the chemostat and exclude the plasmid-bearing organism. The results obtained in the present paper may be of interest in biotechnology. In the commercial production process of genetically altered organisms, in order to avoid capture by the plasmid-free organism, some measures can be taken to reduce the noise intensity so that two microorganisms can coexist to produce the desired products. Now we are in the position to present some numerical examples which will support our analytical results. Using Milstein's Higher Order Method mentioned in [7], we get the following discretization transformation of system (2.1).
By Theorem 5.1, we can conclude that plasmid-bearing organisms will become extinct and plasmid-free organisms will survive, which is supported by Figure 2. Some interesting topics deserve further consideration. Notice that some scholars [4,13] have studied the dynamics of stochastic chemostat models with pulsed input. Next, we will investigate the effects of impulsive perturbations on system (1.2) and find the optimal period of impulsive input. In addition, the chemostat is inevitably affected by temperature, humidity or illumination. At the micro level, the system continuously experiences a transition from one state to another. It is interesting to study model (1.2) with regime switching.