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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.

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

The dynamics of the basic model is simple. If

Note that the modeling process that leads to (

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 [

Taking into account the effect of randomly fluctuating environment, we introduce randomness into model (

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 (

We should point out that a few papers have already addressed 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 Section

Assuming in model (

As

For any given initial value

Since the coefficients of model (

Since the solution is positive, we have

Denote by

By the comparison theorem for stochastic equations, we have

Similarly, we can get

Denote by

and

We have that

To sum up, we have that

Noting that (

The following theorem shows that the solution

For any given initial value

Define the function

Integrating both sides from 0 to

Consequently,

It is clear that

Thus, we complete the proof of Theorem

When

We have known that the dynamic behavior of model (

When

As mentioned above, system (

If

Define a function

Obviously, the function

We will show that for all solutions

Suppose first that

Obviously,

It then follows from (

Suppose next that

Therefore,

It follows from (

To sum up, (

If

The proof is inspired by the method of Imhof and Walcher [

Obviously, the function

We will show that for all solutions

Suppose first that

Obviously,

By (

Suppose next that

By (

Finally, if

By (

To sum up, (

Integrating both sides from

By Strong Law of Large Numbers (see Mao [

It follows from (

This completes the proof of Theorem

Theorems

From the result of Theorem

System (

Let

Obviously, (

Noting also that

It follows from (

Similarly, we have

Therefore, system (

The following theorem shows that sufficiently large noise can make the microorganism extinct exponentially with probability one in a simple chemostat.

For any given initial value

Define the function

Integrating both sides from 0 to

Dividing

The proof of Theorem

In fact, we can see from the proof of Theorem

In order to confirm the results above, we numerically simulate the solution of system (

Deterministic and stochastic trajectories of simple chemostat model for initial condition

Deterministic and stochastic trajectories of simple chemostat model for initial condition

Deterministic and stochastic trajectories of simple chemostat model for initial condition

Deterministic and stochastic trajectories of chemostat model with Holling IV functional response function for initial condition

By Theorem

Next, we consider the effect of stochastic fluctuations of environment on the positive equilibrium of the corresponding deterministic system. As mentioned in the Section

Theorem

To further confirm that Theorem

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.

For the completeness of the paper, we list some basic theory in stochastic differential equations (see [

In general, consider

So

Denote by

If

(i) The trivial solution of

(ii) The trivial solution is said to be stochastically asymptotically stable if it is stochastically stable and, moreover, for every

(iii) The trivial solution is said to be stochastically asymptotically stable in the large if it is stochastically asymptotically stable and, moreover, for all

If there exists a positive-definite decreasing radially unbounded function

The authors are grateful to the handling editor and anonymous referees for their careful reading and constructive suggestions which lead to truly significant improvement of the paper. Research is supported by the National Natural Science Foundation of China (11271260, 11001212, and 1147015) and the Innovation Program of Shanghai Municipal Education Commission (13ZZ116).