Dynamical Analysis of a Stochastic Multispecies Turbidostat Model

A stochastic turbidostat system in which the dilution rate is subject to white noise is investigated in this paper. First of all, sufficient conditions of the competitive exclusion among microorganisms are obtained by employing the techniques of stochastic analysis. Furthermore, the results demonstrate that the competition among microorganisms and stochastic disturbance will affect the dynamical behaviors of microorganisms. Finally, the theoretical results obtained in this contribution are illustrated by numerical simulations.


Introduction
The chemostat and turbidostat, two types of devices for continuous cultivation of microorganism, have been utilized to analyze population dynamics.Novick et al. [1] first proposed the mechanism of chemostat in 1950, and then an increasing number of researchers have devoted themselves to investigate chemostat systems [2][3][4][5][6][7].However, there exist some drawbacks in chemostat model with a constant dilution rate, such as the waste of substrate and higher viscosity caused by the mass transfer efficiency.Models with the dilution rate related to the state of microorganism, which can be called turbidostat (see [8]), can overcome the above drawbacks and be helpful to design optimal strategies to improve the substrate utilization.Flegr [9] investigated turbidostat system with two species from numerical simulation and De Leenheer et al. [4] analyzed the system theoretically.Li [10] systematically investigated the turbidostat model and established sufficient conditions of coexistence of two species.Subsequently, many researchers investigated turbidostat and chemostat systems.Many valuable and interesting results were obtained [11][12][13][14][15][16].
According to May [17], the parameters of the system, such as birth rate, death rate, and the input concentration of nutrient, are inevitably disturbed by environmental factors.
In recent years, more and more stochastic ecological models were chosen to describe the dynamics of populations.Grasmanet et al. [18] established a stochastic chemostat model with three trophic levels.Using singular perturbation methods, they obtained the expected breakdown time and analyzed the influence of stochastic factors on dynamic behaviors of the system in detail.Considering the stochastic factors in chemostat, Campillo et al. [19] proposed stochastic systems with different population scales, and they investigated these systems and derived the field of validity for different population scales.Collet et al. [5] pointed out that the population size in the device was determined not only by the concentration of the nutrient but also by the stochastic birth and death rates.By analyzing the long time behaviors of microbes, they derived the conditions for the global existence of the solutions of the system and the existence of quasi-stationary distribution.Meng et al. [6] constructed an impulsive stochastic chemostat model and investigated the extinction and permanence of the microorganisms.They also pointed out that small perturbation from white noise could cause the extinction of microorganisms.Xu et al. [7] investigated a chemostat model containing telegraph noise with the help of Markov chain and derived the break-even concentration which determines the persistence or extinction 2 Complexity of microorganisms.One can also find a large number of stochastic models on cultivation of microorganisms [14,15,[20][21][22][23] as well as in other fields [24][25][26][27][28].
Competition is all around environment because of the limitation of natural resource.Ghoul et al. [29] and Kaye et al. [30] pointed out that microbes expressed many competitive behaviors with their neighbours or plants for scarce nutrients and limited spaces.Many practical results were obtained by investigating the corresponding competitive models.Butler et al. [16] established a system containing two competitors and a growth-limiting nutrient.They derived conditions for uniform persistence and local stability of the food web and also obtained the conditions for predator-mediated coexistence.Wolkowicz et al. [31] proposed a competitive chemostat model with distributed delay to describe the process of nutrient consumption.They proved that there existed only one survivor under any conditions and also pointed out that the theoretical results in their paper were valid for all systems with monotone growth response functions.Liu et al. [32] established a stochastic competitive model and obtained sufficient conditions of extinction and persistence (including weak persistence and strong persistence).They stated that only one species survived under certain stochastic noise perturbation.Xu et al. [33] analyzed a competitive chemostat model and derived the critical value of the noise.Zhao et al. [34] discussed on a stochastic competition system and derived sufficient conditions of persistence and stationary distribution for each population.Actually, a great number of stochastic competition models have been studied [35][36][37][38][39][40].
In this paper, considering the dilution rate of microorganisms related to the feedback control and the influence of stochastic factors from the environment, we establish the following stochastic turbidostat system based on Zhang et al. [21] who constructed a stochastic chemostat model and obtained the conditions of competitive exclusion.
This paper is organized as follows.In Section 2, the existence and uniqueness of the positive solution of system (1) are verified.Section 3 demonstrates the main results.A discussion is given and a numerical example is offered to verify our theoretical results in Section 4.

The Principle of Competitive Exclusion
In this section, we prove that stochastic turbidostat system (1) satisfies the principle of competitive exclusion.For simplicity, we first define Lemma 3. The solution ((),  1 (), . . .,   ()) of system (1) satisfies with any initial value ((0),  1 (0), . . .,   (0)) ∈  +1 + .Here,  is a positive constant satisfying Proof.Let and where  is a nonnegative constant decided later.By the Itô formula, we derive We can further obtain Denote where Let For 0 <  < , we have where Define a function () by and  fl sup which gives Integrating from 0 to  and taking expectation for (25), one has Then it is easy to obtain which leads to lim () is a continuous function, so there is a constant  > 0 such that According to (24), for small enough  > 0,  = 1, 2, . .., integrating from  to ( + 1) and taking expectation, we can get where here, Applying Burkholder-Davis-Gundy inequality [41], we have Hence, we have Particularly, one can choose  > 0 such that which yields For any  > 0, the following inequality can be obtained by Chebyshev's inequality [41], Using Borel-Cantelli Lemma [41], for all  ∈ Ω and sufficiently large , we have sup Therefore, there is a  0 (), for all  ∈ Ω and  ≥  0 ; the above equation holds.In addition, for all  ∈ Ω, when  ≥  0 and  ≤  ≤ ( + 1), When  → 0, where  is determined by (22) This completes the proof.
where  1 is the break-even concentration of  1 () and  * 1 is the equilibrium concentration of  1 ().
has a solution () weakly converging to the distribution V.
Proof.The deterministic system corresponding to (1) has an equilibrium ( 1 ,  * 1 , 0, . . ., 0) under the condition of  1 <  0 , where  1 and  * 1 satisfy Define  2 function  : Then by the Itô formula, we get where According to (66), we have , It follows from some calculations that Set then where Integrating (68) from 0 to  and dividing by  on both sides, one can obtain Using the Itô formula, we have Then integrating (78) from 0 to  on both sides and dividing all by , we can get Complexity which means ln Based on (76), lim →∞ ( +1 ()/) = 0 and lim →∞ (ln   (0)/ ) = 0, we know that lim ( = 2, . . ., ) . (81) If that is,   () ( = 2, . . ., ) tends to extinction exponentially almost surely.On the basis of classical comparison theorem of stochastic differential equation, the solution () of system (1) and the solution () of system (62) satisfy () ≤ ().According to the Itô formula, it follows that Integrating the above two equations from 0 to  on both sides and dividing them by , we obtain Therefore, which implies that Next, applying the Itô formula to ln  1 (), one deduces Integrating the above inequality from 0 to  on both sides and dividing all by , we have According to Lemma 3, we have which yields that is, Therefore, if This completes the proof.
Proof.It is obvious that () ≤ () by the classical comparison theorem of stochastic differential equation, where () and () are the solutions of system ( 1) and (62), separately.

Discussion and Numerical Simulation
In this paper, a turbidostat model subjected to competition and stochastic perturbation has been investigated.For system (1), we initially proved the existence and uniqueness of global positive solution in Section 2. We further analyzed the principle of competitive exclusion for system (1).Under the conditions of Theorem 6, the species   ( = 2, . . ., ) in the vessel will be extinct and  1 will survive due to the fact that Besides the above assertion, we also obtain the condition for the extinction of system (1), which means that the principle of competitive exclusion becomes invalid if the conditions of Theorem 7 are satisfied.In addition, we have the following remark: Remark 8. Any microorganism population   () ( = 1, 2, . . ., ) in system (1) survives and the other microorganism populations in the vessel will be extinct if the following general condition holds: For a better understanding to the results obtained in Theorem 6, we offer an example as follows.
Although the concentration of nutrients varies significantly in the system, it always fluctuates around  1 = 0.3102 (the black line in ).This fluctuation phenomenon basically comes from the random factors which leads to variation of concentration for () and   ().The species  2 () will be extinct since the concentration of nutrient  * =  1 = 0.3102 is less than its break-even concentration  2 = 0.7128 (see Figure 1).
To further confirm the results obtained in Theorem 7, we give the following example.
In fact, if the density of white noise  +1 increased (such as extremely high temperature) so that (1/  )[  + ( 1 +  2 ) +  2 +1 /2] >  0 ; that is, the environment is not suitable for microbial survival, both adequate and inadequate competitors will be extinct in the end (see Figure 6).
To sum up, the competition from interspecies and stochastic factors from environment may affect the dynamical behaviors of species.Inevitably, some species become adequate competitors and survive after a long time, and the other species in the system become inadequate competitors and die out in the end.However, adequate and inadequate competitors may be extinct under strong enough stochastic disturbance.