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

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.


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][2][3][4][5][6][7][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] 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][11][12][13][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][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][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][37][38][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   ( = 1, 2) is perturbed by white noises so that where   () is a standard Brownian motion with intensity   > 0. 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.

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 (Ω, F, {F} ≥0 , P) be a complete probability space with a filtration {F  } ≥0 satisfying the usual conditions: it is increasing and right continuous while F 0 contains all P-null sets; we use () to represent a scalar Brownian motion defined on the complete probability space Ω; also let  2 + = {  > 0,  = 1, 2}.If for an integrable function  on [0, +∞), define Then we have the following.
Then, one can show the following lemmas.

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  0 ( 0 1 ,  0 2 , 0).Let  * ( * 1 ,  * 2 ,  * ) be the coexistence equilibrium of model (2), which satisfies where Then we have that where Thus, equation has one positive root  2 at least, and  2 ∈ (0,  0 2 ).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  1 at least, and  1 ∈ (0,  0 1 ).From the third equation of ( 12), we have 0 <  <  0 1 +  0 2 .Then we have the following theorem.Regarding the stability of these equilibria, we have the following theorem.Theorem 5. Then for system (2), one has the following.
Proof.Linearizing the system at the equilibrium where The characteristic equation gives where Obviously, we have, at  0 , Then we have and thus if R < 1, all the eigenvalues of ( 23) have negative real part; then, by the stability theory,  0 is stable.And, at  * , we have here  3 +  4 =  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.

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 R is introduced in ( 16).Then we have the following.
Theorem 6.For system (5), if one of the following holds, , and R * < 1, then the microorganism () of system (5) goes to extinction almost surely.Moreover, almost surely.
The same discussion can be used in Case 3; here we omit it.
Next, we consider Case 4: Dividing both sides of (42) by  > 0, we have and, by Lemma 3, we have Then, taking the limit superior on both sides of (43) leads to lim sup which implies lim →+∞ () = 0.

Mathematical Problems in Engineering
Next, we prove the last conclusion.Given 0 <  ≪ 1, since lim →+∞ () = 0, we have 0 < () <  for  large enough.By the first equation of system (5), we have Then when  → 0 we have lim inf On the other hand from the proof of Lemma 2, we have Let  → 0. Then one has lim sup From ( 47) and ( 49 almost surely.This completes the proof of Theorem 6.
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.

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.
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  1 and  2 and discuss below five different cases.