Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 5217027, 11 pages
https://doi.org/10.1155/2017/5217027

Research Article

Threshold Dynamics of a Stochastic Chemostat Model with Two Nutrients and One Microorganism

Jian Zhang,¹ Anqi Miao,¹ and Tongqian Zhang1,2

¹College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
²State Key Laboratory of Mining Disaster Prevention and Control Co-Founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China

Correspondence should be addressed to Tongqian Zhang; zhangtongqian@sdust.edu.cn

Received 30 March 2017; Revised 15 July 2017; Accepted 20 August 2017; Published 25 September 2017

Academic Editor: Zhongwei Lin

Copyright © 2017 Jian Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

A new stochastic chemostat model with two substitutable nutrients and one microorganism is proposed and investigated. Firstly, for the corresponding deterministic model, the threshold for extinction and permanence of the microorganism is obtained by analyzing the stability of the equilibria. Then, for the stochastic model, the threshold of the stochastic chemostat for extinction and permanence of the microorganism is explored. Difference of the threshold of the deterministic model and the stochastic model shows that a large stochastic disturbance can affect the persistence of the microorganism and is harmful to the cultivation of the microorganism. To illustrate this phenomenon, we give some computer simulations with different intensity of stochastic noise disturbance.

1. Introduction

Chemostat is commonly used to describe the dynamics of a microbial population in a continuous bioreactor in which microorganisms grow on a substrate and has attracted great interest of many scholars [1–8], since it was first introduced by Monod [9]. A single simple species chemostat model with Michaelis-Menten-Monod functional response was proposed by [9] as follows:

where is the concentration of the nutrient, is the concentration of the organism, is the dilution (or washout) rate, is the maximal growth rate, is the Michaelis-Menten (or half-saturation) constant with units of concentration, and is a “yield” constant reflecting the conversion of nutrient to organism.

However, experimental results have indicated that the microorganisms depend on a variety of nutrition substances such as carbon, nitrogen, energy, growth factors, inorganic salts, and water. Then the model of microorganisms species growth in the chemostat on two nutrients is considered by [10–14]. A model of single-species growth in the chemostat on two substitutable resources with Michaelis-Menten-Monod functional response was proposed by [14] as follows:

However, it is now well known that stochastic noise is widely present in biological systems and so on [15–33] and microorganisms are inevitably influenced by some random factors in the process of cultivation. To better understand the dynamic behavior of the chemostat, a host of scholars proposed a slice of stochastic chemostat models and studied the effect of the random noise on the dynamic behavior of the stochastic models. As an example, Imhof and Walcher [34] proposed a stochastic chemostat model for a single microorganism species consuming a single nutrient. They found that random effects may lead to extinction in scenarios where the deterministic model predicts persistence. Recently, Xu and Yuan [35] established a stochastic chemostat model in which the maximal growth rate is influenced by the white noise in environment as follows:

They got an analogue break-even concentration involving the white noise which can determine the exclusion and persistence of the microorganism. And more stochastic chemostat models can be found in [36–39].

Motivated by the papers mentioned above, in this paper, we further consider a model of single-species growth in the chemostat on two supplementary resources with Michaelis-Menten-Monod functional response and environmental noise. We assume that the maximal growth rate is perturbed by white noises so that

where is a standard Brownian motion with intensity Then the resultant model takes the following form:

Our main objective in the rest of this paper is to investigate the threshold dynamics of stochastic chemostat model (5) and explore the conditions under which microorganisms will die out or exist.

2. Preliminaries

In this section, we will give some notations, definitions, and lemmas which will be used for analyzing our main results. To this end, throughout this paper, we let be a complete probability space with a filtration satisfying the usual conditions: it is increasing and right continuous while contains all -null sets; we use to represent a scalar Brownian motion defined on the complete probability space ; also let . If for an integrable function on , define

Then we have the following.

Definition 1. For system (5),

(i)the microorganism is said to be extinctive if ,(ii)the microorganism is said to be permanent in mean if there exists a positive constant such that .

Then, one can show the following lemmas.

Lemma 2. The solution of model (2) or (5) with the initial condition is ultimately bounded; that is,

where

Proof. Letting , from system (2) or system (5), we have

This implies that

Thus, we have

This completes the proof of Lemma 2.

By Lemma 2 and the strong law of large numbers for martingales [40], we can obtain the following lemma.

Lemma 3. Letting be a solution of system (5) with initial value , then

3. Dynamics of Deterministic System (2)

In this section, we will focus on the deterministic system (2). It is easy to see that the equilibria point of (2) satisfy

and, obviously, model (2) has a microorganism extinction equilibrium Let be the coexistence equilibrium of model (2), which satisfies

where

Then we have that

Denote

where ,

Obviously, If , we have If , we have Thus, equation has one positive root at least, and

From the second equation of (12), one gets

Substituting (17) into the first equation of (12), we have

Let

It is easy to see that

Thus, (18) has one positive root at least, and

From the third equation of (12), we have Then we have the following theorem.

Theorem 4. If and , then system (2) has unique positive equilibrium

Regarding the stability of these equilibria, we have the following theorem.

Theorem 5. Then for system (2), one has the following.

(i)If , microorganism extinction equilibrium is locally stable; if it is unstable.(ii)If and , the coexistence equilibrium is locally stable.

Proof. Linearizing the system at the equilibrium gives the Jacobian

where

The characteristic equation gives

where

Obviously, we have, at ,

Then we have

and thus if , all the eigenvalues of (23) have negative real part; then, by the stability theory, is stable.
And, at , we have

here is used. Then all the eigenvalues of (23) have negative real part; thus, by the stability theory, the diseases equilibrium is stable as long as it exists.

4. Dynamics of Stochastic System (5)

4.1. Extinction

In this section, we explore the conditions leading to the extinction of the two infectious diseases. Denote

where is introduced in (16). Then we have the following.

Theorem 6. For system (5), if one of the following holds,

(i), , and ,(ii), , and ,(iii), , and ,(iv), , and ,

then the microorganism of system (5) goes to extinction almost surely. Moreover,

almost surely.

Proof. Let be a solution of system (5) with initial value . Applying Itô’s formula to system (5) results in

where ,
Integrating both sides of (30) from to gives

where

known as the local continuous martingale, and . Obviously, we need to estimate the maximum value of
Let us consider quadratic function

It is easy to verify that when , reaches its maximum value at ; and when , achieve its maximum value at Then, in (31), we have four cases to be discussed, depending on whether or , which are as follows: Case 1: , ; Case 2: , ; Case 3: , ; and Case 4: ,
For Case 1, since , , then achieve the maximum value Then we can easily see from (31) that

Dividing both sides of (34) by , we have

and, by Lemma 3, we have

Then, taking the limit superior on both sides of (35) leads to

which implies , and here is used.
Case 2. ,
In this case, we can easily see from (31) that

Dividing both sides of (38) by , we have

and, by Lemma 3, we have

Then, taking the limit superior on both sides of (38) leads to

which implies
The same discussion can be used in Case 3; here we omit it.
Next, we consider Case 4: , From (31), we have

Dividing both sides of (42) by , we have

and, by Lemma 3, we have

Then, taking the limit superior on both sides of (43) leads to

which implies
Next, we prove the last conclusion. Given , since , we have for large enough. By the first equation of system (5), we have

Then when we have

On the other hand from the proof of Lemma 2, we have

Let . Then one has

From (47) and (49), we have

almost surely.
By employing the method similar above, it then follows that

almost surely. This completes the proof of Theorem 6.

4.2. Permanence in Mean

Theorem 7. If , then the microorganism is permanent in mean; moreover, satisfies

where

Proof. Integrating from to and dividing by on both sides of system (5) yield

Then one can get

Applying Itô’s formula gives

Integrating from 0 to and dividing by on both sides of (55) yields

where , Noticing that

then we have

If , we can get

By inequality (59), we have

By Lemma 3, we get that According to Lemma 2, one sees that , , and , and then one has and Thus taking the inferior limit of both sides of (60) yields

And if , we can get

where ,
Inequality (62) can be rewritten as

Taking the inferior limit of both sides of (63) yields

Let , and we get from (61) and (64)

This completes the proof of Theorem 7.

Remark 8. Theorems 6 and 7 show that the condition for the microorganism to go to extinction or permanence depends on the intensity of the noise disturbances completely. And small noise disturbances will be beneficial to the cultivation of the microorganism; conversely, large white noise disturbance is harmful to the cultivation of the microorganism.

5. Conclusion and Numerical Simulation

This paper proposes and investigates a new stochastic chemostat model with two substitutable nutrients and one microorganism. Then main objective in this paper is to investigate the threshold dynamics of stochastic chemostat model (5) and explore the conditions which can determine the extinction and permanence of the microorganism using two substitutable nutrients. Firstly, for the corresponding deterministic model, the threshold for extinction or existence of the microorganism is obtained by analyzing the stability of the equilibria. Then the threshold of the stochastic chemostat for the extinction and the permanence in mean of the microorganism is explored. The results show that there exists a significant difference between the threshold of the deterministic system and the stochastic system, which makes the persistent microorganism of a deterministic system become extinct due to large stochastic disturbance. That is, large stochastic disturbance is harmful to the cultivation of the microorganism. It is worth mentioning that this paper is a promotion of the work of Xu and Yuan [35].

Next, using the Euler Maruyama (EM) method [40], we give some numerical simulation to illustrate the extinction and persistence of the microorganism in stochastic system and corresponding deterministic system for comparison.

Firstly, we begin from a deterministic system; the basic parameters are set as , , , , , , and Direct calculation shows that , , and Then according to Theorems 4 and 5, the deterministic system has a unique stable positive equilibrium , which is locally stable and the deterministic system is permanent (see Figure 1).

Figure 1: Time series for the paths , , for the deterministic system with , , , , , , ,

Next, we consider the influence of stochastic disturbance on the above deterministic system. According to Theorem 6, different parameters are chosen to give insights into the reasonability of the results stated in Theorem 6.

We choose different value of parameters and and discuss below five different cases.

Case 1. Choose , , by direct calculation; we have Then, by Theorem 6, the microorganism eventually tends to be extinct (see Figure 2(a)).

(a),

(b),

(c),

(d), ,

Figure 2: Time series for the paths , , for the stochastic system.

Case 2. Choose , , by direct calculation; we have Then, by Theorem 6, the microorganism eventually tends to be extinct (see Figure 2(b)).

Case 3. Choose , , by direct calculation; we have Then, by Theorem 6, the microorganism eventually tends to be extinct (see Figure 2(c)).

Case 4. Choose , , by direct calculation; we have Then, by Theorem 6, the microorganism eventually tends to be extinct (see Figure 2(d)).

Case 5. Choose , , by direct calculation; we have Then, by Theorem 7, the microorganism is persistent (see Figure 3).

Figure 3: Time series for the paths , , for the stochastic system with , ,

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is supported by Shandong Provincial Natural Science Foundation (no. ZR2015AQ001), the National Natural Science Foundation of China (no. 11371230), and Research Funds for Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources by Shandong Province and SDUST (2014TDJH102).

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