Dynamical Analysis of Two-Microorganism and Single Nutrient Stochastic Chemostat Model with Monod-Haldane Response Function

Chi, Mengnan; Zhao, Wencai

doi:https://doi.org/10.1155/2019/8719067

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Research Article | Open Access

Volume 2019 | Article ID 8719067 | https://doi.org/10.1155/2019/8719067

Dynamical Analysis of Two-Microorganism and Single Nutrient Stochastic Chemostat Model with Monod-Haldane Response Function

Mengnan Chi¹and Wencai Zhao^1,2

Academic Editor: Thach Ngoc Dinh

Received30 Oct 2018

Revised15 Feb 2019

Accepted21 Feb 2019

Published10 Mar 2019

Abstract

In this paper, we formulate and investigate a two-microorganism and single nutrient chemostat model with Monod-Haldane response function and random perturbation. First, for the corresponding deterministic system, we introduce the conditions of the stability of the equilibrium points. Then, using Lyapunov function and Itô’s formula, we investigate the existence and uniqueness of the global positive solution of the stochastic chemostat model. Furthermore, we explore and obtain the criterions of the extinction and the permanence for the stochastic model. Finally, numerical simulations are carried out to illustrate our main results.

1. Introduction

The chemostat is a classic bioreactor for continuous microbial culture. It has been widely used in microbiology and bioengineering [1–3]. The device is mainly composed of three parts: nutrient bottle, incubator, and collector, generally by using pump or an overflow device to keep the chemostat volume constant. Chemostat model has attracted great interests of many scholars since Monod [4]. Monod [5], Novick et al. [6], and Herbert et al. [7] considered the basic theory of microbial interaction in the chemostat. The simple chemostat model described by ordinary differential equations (ODEs) takes the following form: where and stand for the concentrations of the nutrient and the microorganism at time , respectively. and are positive constants, which represent the input concentration of the nutrient and the common washout rate respectively. The function denotes the Monod growth functional response, is called the maximal growth rate, and is the half-saturation (or Michaelis-Menten) constant. Based on the above model, many scholars have studied the chemostat model with different functional response in which a single microbe feeds on a single nutrient [8–11].

Taylor and Williams [12] considered the coexistence of two different microbe populations feeding on single nutrient in the chemostat and proposed the following model: where and denote the concentrations of two different microorganisms at time And then many scholars have investigated the competition of microorganisms in chemostat [13–16].

Taking into account more complex inhibitory effects of high substrate concentrations on microbial growth, for example, nitrite and ammonia can lead to the inhibition of Nitrobacter and Nitrosomonas, respectively. Andrews [17] proposed the following nonmonotonic response function called Monod-Haldane growth rate (inhibitory rate):where is the maximal growth rate and is the Michaelis-Menten constant, assuming that the term is an inhibitor and is a half-saturation parameter. Based on [17], Bush and Cook [18] improved the inhibition function to a general functional response which retains the particular features of the Andrews function. Recently, Wang et al. [19] proposed and investigated the stochastic dynamic behaviors of a stochastic chemostat model with Monod-Haldane response function in which a single microbe feeds on a single nutrient.

Considering the nutritional substitution, Chi and Zhao [20] established a single microorganism and multinutrient chemostat model with impulsive toxicant input in a polluted environment as follows: In the above model, a single species () feeds on two substitutable resources ( and ) in a polluted environment. For the system without stochastic effect, by using the theory of impulsive differential equations, the authors proved that the model system has a globally asymptotically stable ‘microorganism-extinction’ periodic solution for and the system is permanent for And for the system with stochastic effect, by using the theory of stochastic differential equations, the authors obtained the conditions for the persistence and extinction of microorganisms. Their results showed that stochastic disturbance and toxicant can affect the survival of microorganism.

While, in reality, there are generally various microorganisms coexisting in lakes and oceans, which might depend on single nutrient in the region. Different from the model in Chi and Zhao [20], in present paper, we consider two different microbes to compete for a nutrient, by introducing Monod-Haldane functional response; we get the model as follows:where and denote the intraspecies competition rates. The meanings of other parameters are the same as those described above.

It is well known that nature is often affected by random factors [21–25], unavoidably the process of microbial culture is affected by the interference of random factors [26–28], such as the degradation of microbial strains, the existence of an inducer or inhibitor on the growth, cultivation temperature changes, and the species and concentrations of inorganic salts changes. To understand the phenomenon of stochastic perturbations deeply [29–31], many scholars have studied the effect of the noise on the dynamical behavior of the stochastic chemostat models [32–37].

A parameter of the system is often subject to random disturbance [38–42]; thus, in this paper, we assume that the maximal growth rates are perturbed by environment noise on the basis of the approaches used in [20, 26, 43]. In this case, change to random variables , and , where are independent standard Brownian motions with intensity . In summary, we replace in deterministic model (5) with to get the following stochastic differential equations model:

The remaining part of this paper is organized as follows. In Section 2, we give some notations, definitions, and lemmas which will be used in the following section. In Section 3, we explore the sufficient conditions of system (6) for the extinction and persistence of the two microorganisms. Finally, some conclusions and numerical simulations are given in Section 4.

2. Preliminary Knowledge

In this paper, we denote For an integrable function on , define

Definition 1. (i) The microorganism and are said to be extinctive if and .
(ii) The microorganism and are said to be permanent in mean if there exist two positive constants and such that and .

Lemma 2. For any positive solution of systems (5) or (6) with initial value , we have

Proof. From system (5) or (6), we haveThis implies thatThen obviously we obtainThis completes the proof of Lemma 2.

Lemma 3. (a) System (5) has at most three equilibrium points on the boundary of the positive quadrant. The trivial equilibrium point always exists. It is a locally asymptotically stable ‘microorganism-extinction’ equilibrium point when and
(b) If , then there exists a equilibrium point with , and and satisfy the equation . It is a locally asymptotically stable equilibrium point when and .
(c) If , then there exists a equilibrium point with , and and satisfy the equation . It is a locally asymptotically stable equilibrium point when and

Proof. (I) Firstly, we prove the existence of boundary equilibrium point. In system (5), let By the above formula, we get the following equilibria: Let and ; we get the following equation through system (5):Let ; then, by simplifying (13) we getFor system (14), if , we can see that the two functional images have unique intersection point, which proves that system (13) has a positive solution and satisfies the equation . So it proves the existence of .
Similarly, we can prove the existence of
(II) Secondly, we prove the stability of the boundary equilibrium point. The Jacobian matrix associating to the equilibrium of system (5) is where (i) The stability of the ‘microorganism-extinction’ equilibrium point of system (5) is determined by the Jacobian which has the following eigenvaluesAccording to stability theory [44, 45], is stable if and ; i.e., and .
(ii) The functional matrix at is equal to which has the following eigenvalue:The other eigenvalues satisfy the following equations:whereIf , then and . Hence, it follows that any root of (21) has negative real part owing to the Routh-Hurwitz criterion. According to stability theory [44], is stable if and .
Similarly, we can prove the stability of .
This finishes the proof of Lemma 3.

Remark 4. We also can obtain the conditions and for the existence of equilibrium points and from the Proposition 1 and Proposition 2 in [34].

Lemma 5. For any initial value , system (6) has a unique solution on , and the solution remains in with probability one; namely, for all almost surely.

Proof. Since the coefficients of system (6) are locally Lipschitz continuous, for any given initial value , there is a unique local solution on , where is the explosion time. If we can prove a.s., then the solution will be global.
First we set and is an empty set. Select such that For any positive , define the stopping time as follows:Clearly is increasing as Assuming , obviously We declare that holds a.s.; thus, a.s. and for all So we only need to prove a.s.
If this assertion is false, then there is a pair of constants and such that . Hence, there is a positive constant such that for any positive
From system (6), the total biomass satisfies the equation then leads to that for all ,Define a -function : byObviously, is positive defined. Applying Itô formula, we obtainwhere So we getIntegrating both sides from 0 to , and then taking expectations, we haveDenote and for any positive . Note that for every , at least, one of or equals to , and thenSo we obtain where is the indicator function of . Letting leads to the contradictionSo almost surely. The proof is complete.

3. Extinction and Persistence of System (6)

3.1. Extinction

In this section, we will try to give conditions that lead to the extinction of microorganism. LetThen we get the following theorem.

Theorem 6. Let be a solution of system (6) with initial value Then if one of the following holds and , or and , or and , or and , the microorganism goes to extinction almost surely, i.e., , a.s.

Proof. Applying Itô’s formula to system (6) yieldsIntegrating from 0 to and dividing by on both sides of (34) yields where the function By the strong law of large number and Lemma 2, we getThen four cases should be discussed.
since and , then we can easily see from (35) thatTaking the superior limit on both sides of (37) yieldswhich implies , a.s.
Similarly, we can prove that microorganism also tends to extinction under condition , and we omit it here.
and From (35), we haveTaking the superior limit on both sides of (39) leads towhich implies , a.s.
and In this case, we can see from (35) thatTaking the superior limit on both sides of (41) yieldswhich implies , a.s.
This completes the proof of Theorem 6.

The same discussion can be used in ; we have the following.

Theorem 7. Let be a solution of system (6) with initial value Then if one of the following holds and , or and , or and , or and , the microorganism goes to extinction almost surely, i.e., , a.s.

3.2. Permanence in Mean

For system (6), letand Then we get the following theorem.

Theorem 8. Let be a solution of system (6) with initial value , then we have the following.
(i) If , satisfies one of the conditions , and of Theorem 7, then the microorganism goes extinct and the microorganism is permanent in mean; moreover, satisfies (ii) If , satisfies one of the conditions , and of Theorem 6, then the microorganism goes extinct and the microorganism is permanent in mean; moreover, satisfies (iii) If and , then two microorganisms and are permanent in mean; moreover, and satisfy where

Proof. Case (i): by Theorem 7, since satisfies one of the conditions , and of Theorem 7, then Since , for small enough, such that for all large enough andIntegrating from 0 to and dividing by on both sides of system (6) yields then one can get Applying Itô’s formula givesIntegrating from 0 to and dividing by on both sides of (52) yieldswhere The inequality (53) can be rewritten as By the strong law of large numbers and Lemma 2, we have and Taking the inferior limit of both sides of (54) and using Lemma 2.5 in [46], we can get Let , then we haveThe same discussion can be used in Case (ii), and we omit it here.
Case (iii): notice thatDefinewhere and . Then we have Integrating from 0 to and dividing by on both sides of (59) yields where The inequality (60) can be rewritten as By the strong law of large numbers and Lemma 2, we have and Taking the inferior limit of both sides of (61) yieldsThis proof is completed.

4. Conclusions and Simulations

In this paper, we propose and analyse the dynamics of two-microorganism and single nutrient stochastic chemostat models with Monod-Haldane response function. First, we discuss the existence and locally asymptotical stability of boundary equilibria of the system neglecting stochastic effect. Then, we investigate the dynamics of the system under stochastic effect and obtain the conditions which determine the persistence and extinction of the microorganisms with stochastic effect. Our results show that large stochastic noise can lead to microbial extinction (see Theorems 6 and 7), and small stochastic noise is beneficial to the survival of microorganisms (see Theorem 8).

In order to verify the theoretical results obtained in this paper, we give some numerical simulation. We choose the parameters in model (5) and model (6) as follows:

Note that

Theorems 6 and 7 show that the microorganism die out under a large white noise disturbance intensity (see Figure 1(a) with ). Figure 1 shows that the persistent microorganism of a deterministic system (see Figure 1(b)) can become extinct due to the white noise stochastic disturbance. Therefore, the large white noise stochastic disturbance intensity is detrimental to the survival of microorganisms.

(a)

(b)

Theorem 8 indicates that the microorganism can become extinct or persistent under a small white noise disturbance intensity. We keep the system parameters same as those in Figure 1, and let and take different parameter values. When is larger and is smaller (), here . Thus, the microorganism tends to die out, and the microorganism is persistent (see Figure 2(a)). Conversely, when is smaller and is larger (), here , Figure 2(b) shows that the microorganism tends to die out, and the microorganism is persistent. Furthermore, let and take small parameter values (); here , then both and are persistent (see Figure 2(c)); that is, the two microorganisms can coexist. Although and still take small parameter values (), when , both and can become extinct (see Figure 2(d)). The above simulations support our results in the article.

(a)

(b)

(c)

(d)

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (11371230) and SDUST Research Fund (2014TDJH102).

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Copyright © 2019 Mengnan Chi and Wencai Zhao. 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.

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