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

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.


Introduction
The chemostat is a classic bioreactor for continuous microbial culture.It has been widely used in microbiology and bioengineering [1][2][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. 0 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][9][10][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 2 Complexity investigated the competition of microorganisms in chemostat [13][14][15][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):  ( ()) =  ()  +  () +  2 () , where  > 0 is the maximal growth rate and  > 0 is the Michaelis-Menten constant, assuming that the term  2 () 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 ( 1 () and  2 ()) 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  1 < 1 and the system is permanent for  2 > 1.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  1 and  2 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][22][23][24][25], unavoidably the process of microbial culture is affected by the interference of random factors [26][27][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][30][31], many scholars have studied the effect of the noise on the dynamical behavior of the stochastic chemostat models [32][33][34][35][36][37].
A parameter of the system is often subject to random disturbance [38][39][40][41][42]; thus, in this paper, we assume that the maximal growth rates   ( = 1, 2) are perturbed by environment noise on the basis of the approaches used in [20,26,43].In this case,   ( = 1, 2) change to random variables m ( = 1, 2), and m =   +   Ḃ  ()( = 1, 2), where   ()( = 1, 2) are independent standard Brownian motions with intensity   ≥ 0( = 1, 2).In summary, we replace   ( = 1, 2) in deterministic model (5) with   +   Ḃ  ()( = 1, 2) 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.
(ii) The functional matrix at ( 0 ,  1 , 0) is equal to which has the following eigenvalue: The other eigenvalues satisfy the following equations: where Hence, it follows that any root of (21) has negative real part owing to the Routh-Hurwitz criterion.According to stability theory [44], Similarly, we can prove the stability of  2 ( 0 , 0,  2 ).This finishes the proof of Lemma 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.Let Then we get the following theorem.
Proof.Applying Itô's formula to system (6) yields Integrating from 0 to  and dividing by  on both sides of (34) yields where the function Then four cases should be discussed.
The same discussion can be used in (); we have the following.

Permanence in Mean.
For system (6), let and Then we get the following theorem.

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: Theorems 6 and 7 show that the microorganism die out under a large white noise disturbance intensity (see Figure 1(a) with  1 = 1.7, 2 = 1.9).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.