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) satisfyand, obviously, model (2) has a microorganism extinction equilibrium Let be the coexistence equilibrium of model (2), which satisfieswhere Then we have that Denotewhere ,
Obviously, If , we have If , we have Thus, equation has one positive root at least, and
From the second equation of (12), one getsSubstituting (17) into the first equation of (12), we haveLet 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 Jacobianwhere The characteristic equation giveswhere 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 inwhere ,
Integrating both sides of (30) from to giveswhere known as the local continuous martingale, and . Obviously, we need to estimate the maximum value of
Let us consider quadratic functionIt 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) thatDividing both sides of (34) by , we haveand, 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) thatDividing both sides of (38) by , we haveand, 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 haveDividing both sides of (42) by , we haveand, 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 haveOn the other hand from the proof of Lemma 2, we have Let . Then one hasFrom (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) yieldThen one can getApplying Itô’s formula givesIntegrating from 0 to and dividing by on both sides of (55) yields where , Noticing that then we haveIf , we can get By inequality (59), we haveBy 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) yieldsAnd if , we can get where ,
Inequality (62) can be rewritten asTaking the inferior limit of both sides of (63) yieldsLet , 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).
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) , ,
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).
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).