# Dynamical Analysis of a Stochastic Multispecies Turbidostat Model

Authors: Yu Mu; Zuxiong Li; Huili Xiang; Hailing Wang
Journal: Complexity
 (2019)

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## Abstract

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.


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## Body

## 1. 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 [2–7]. 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 [11–16].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 of microorganisms. One can also find a large number of stochastic models on cultivation of microorganisms [14, 15, 20–23] as well as in other fields [24–28].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 [35–40].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.(1)dSt=S0-Std+∑i=1nkixit-∑i=1naiSt1+biStxitdt+σ1StdB1t,dxit=-di+∑i=1nkixitxit+aiSt1+biStxitdt+σi+1xitdBi+1t,i=1,2,…,n,where S(t) and xi(t)(i=1,2,…,n) represent the concentration of nutrient and microorganisms at the time t, separately. S0 expresses the input concentration of nutrition, Bi(t)(i=1,2,…,n+1) is Brownian motion defined on a complete probability space (Ω,F,P) (Ft=σ{(S(t),x1(t),x2(t),…,xn(t));0≤t≤τe}) and σi>0(i=1,2,…,n+1) stands for the density of the white noise, d+∑i=1nkixi (d>0 and ki>0 are constants) denotes the dilution rate of the turbidostat and di+∑i=1nkixi (di>0 is constant) is the sum of the dilution rate and death rate of the microorganisms in the system. aiS/1+biS is the Holling 2 functional response function (ai>0 and bi≥0 are constants).This paper is organized as follows. In Section2, 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.


## 2. Existence and Uniqueness of the Positive Solution

We in this section demonstrate the existence and uniqueness of global positive solution of system (1).Theorem 1.
Ifdi/ki>S0+1(i=1,2,…,n), then stochastic system (1) has a unique positive solution (S(t),x1(t),…,xn(t)) almost surely with initial value (S(0),x1(0),…,xn(0))∈R+n+1.Proof.
Fort≥0, let(2)ut=ln⁡St,vit=ln⁡xit,i=1,2,…,n.According to the Ito^ formula, we have(3)dut=S0-eutd+∑i=1nkieviteut-∑i=1naievit1+bieut-σ122dt+σ1dB1t,dvit=-di+∑i=1nkievit+aieut1+bieut-σi+122dt+σi+1dBi+1t,i=1,2,…,n,with initial value (u(0),v1(0),…,vn(0))=(ln⁡S0,ln⁡x10,…,ln⁡xn(0)). It is obvious that eu and evi are continuous and differentiable functions. Hence (3) satisfies local Lipschitz condition, which means that there exists a unique local solution (u(t),v1(t),…,vn(t)) for t∈[0,τe). Then (S(t),x1(t),…,xn(t))=(eu(t),ev1(t),…,evn(t)) is the unique positive local solution for system (1) with the initial value (S(0),x1(0),…,xn(0))∈R+n+1. Next, we prove that the positive solution of system (1) is global; that is, τe=∞. We first define the filtration {Ft} as Ft={Xs;0≤s≤τe} and choose an enough large k0>0 such that (S(0),x1(0),…,xn(0))∈[1/k0,k0]. Then for all k≥k0, we define a stopping time by(4)τk=inf⁡t∈0,τe:min⁡St,x1t,…,xnt≤1kormax⁡St,x1t,…,xnt≥k.
Next, we further showτe→∞. According to the definition of the stopping time, (5)min⁡St,x1t,…,xnt≤1k→0,max⁡St,x1t,…,xnt≥k→∞,ask→∞.Therefore, τk is increasing as k→∞. Set τ∞=limk→∞τk. Then τ∞≤τe. Now, it is sufficient for us to show τ∞=∞. Arguing by contradiction, we assume that τ∞≠∞. Then there exist constants T>0 and ε∈(0,1) such that P(τ∞≤T)>ε. Therefore, there is a constant k1≥k0 such that P(τk≤T)≥ε for all k≥k1.
Define aC2 function V:R+n+1→R¯+ by(6)VS,x1,x2,…,xn=S-a-aln⁡Sa+∑i=1nxi-1-ln⁡x1,where a=min⁡di-S0+1ki/ki+ai(i=1,2,…,n) is a positive constant. It is obvious that u-1-ln⁡u≥0,∀u≥0. Hence V(S,x1,…,xn)≥0. By the Ito^ formula, the following equation can be derived(7)dV=S-aSS0-Sd+∑i=1nkixi-S-aS∑i=1naiS1+biSxi+aσ122-∑i=1nxi-1xidi+∑i=1nkixixi+∑i=1nxi-1xiaiSxi1+biS+∑i=1nσi+122dt+S-aσ1dB1t+∑i=1nxi-1σi+1dBi+1t,where (8)LV=S-aSS0-Sd+∑i=1nkixi-S-aS∑i=1naiS1+biSxi+aσ122-∑i=1nxi-1xidi+∑i=1nkixixi+∑i=1nxi-1xiaiSxi1+biS+∑i=1nσi+122=S0+ad+∑i=1nkixi+∑i=1naaixi1+biS+aσ122+∑i=1ndi+∑i=1nkixi+∑i=1nσi+122-Sd+∑i=1nkixi-aS0Sd+∑i=1nkixi-∑i=1ndi+∑i=1nkixixi-∑i=1naiS1+biS. Through calculations, the following inequality can be obtained (9)LV≤S0+ad+∑i=1nkixi+∑i=1ndi+∑i=1nkixi+aσ122+∑i=1nσi+122+∑i=1naaixi-∑i=1ndixi≤S0+ad+∑i=1ndi+aσ122+∑i=1nσi+122+∑i=1nS0+aki+ki+aai-dixi=S0+ad+∑i=1ndi+aσ122+∑i=1nσi+122+∑i=1nki+aia-di-S0+1kiki+aixi. Defining a=min⁡di-S0+1ki/ki+ai(i=1,2,…,n), we have(10)LV≤S0+ad+∑i=1ndi+aσ122+∑i=1nσi+122, which yields(11)dV=LVdt+S0-aσ1dB1t+∑i=1nσi+1xi-1dBi+1t≤S0+ad+∑i=1ndi+aσ122+∑i=1nσi+122+S0-aσ1dB1t+∑i=1nσi+1xi-1dBi+1t. Integrating from 0 to τk∧T and taking expectation in both sides of inequality (11), one can obtain(12)EVSτk∧T,…,xnτk∧T≤VS0,…,xn0+S0+ad+∑i=1ndi+aσ122+∑i=1nσi+122T.Let Ωk={τk≤T} for all k≥k1, then P(Ωk)≥ε. Based on the definition of the stopping time, we have mi{S(t),x1(t),…,xn(t)}≤1/k or ma{S(t),x1(t),…,xn(t)}≥k for all ω∈Ωk. Hence there is a positive constant k such that V(S(τk∧T),…,xn(τk∧T))≥k. Furthermore, (13)VS0,…,xn0+S0+ad+∑i=1ndi+aσ122+∑i=1nσi+122T≥EVSτk∧T,…,xnτk∧T≥PΩkVSτk∧T,…,xnτk∧T≥εk. Setting k→∞, we have V(S(0),…,xn(0))+((S0+a)d+∑i=1ndi+aσ12/2+∑i=1nσi+12/2)T≥∞, which is contradictory with V(S(0),…,xn(0))<∞. Hence τe=∞. This completes the proof.Remark 2.
If we takeT=di/ki-S0(i=1,2,…,n) as a threshold value, then the microorganisms in system (1) will survive as T>1 and be extinct as T≤1.


## 3. 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(14)xt=1t∫0txrdr.Lemma 3.
The solution(S(t),x1(t),…,xn(t)) of system (1) satisfies(15)limt→∞supln⁡St+∑i=1nxitln⁡t≤1θ,a.s. with any initial value (S(0),x1(0),…,xn(0))∈R+n+1. Here, θ is a positive constant satisfying(16)λ=min⁡d,d1,…,dn-∑i=1nkiS0-θ-12∨0σ12∨…∨σn2>0.Proof.
Let(17)ut=St+∑i=1nxit, and (18)Wu=1+uθ, where θ is a nonnegative constant decided later. By the Ito^ formula, we derive (19)du=S0-Sd+∑i=1nkixi-∑i=1ndi+∑i=1nkixixidt+σ1SdB1t+∑i=1nσi+1xidBi+1t. We can further obtain (20)dW=θ1+uθ-21+uS0-Sd+∑i=1nkixi-∑i=1ndi+∑i=1nkixixi+θ-12σ12S2+∑i=1nσi+12xi2dt+θ1+uθ-1σ1SdB1t+∑i=1nσi+1xidBi+1t. Denote (21)LWu=θ1+uθ-21+uS0-Sd+∑i=1nkixi-∑i=1ndi+∑i=1nkixixi+θ-12σ12S2+∑i=1nσi+12xi2=θ1+uθ-21+uS0d+∑i=1nkixi-Sd+∑i=1nkixi-∑i=1ndi+∑i=1nkixixi+θ-12σ12S2+∑i=1nσi+12xi2≤θ1+uθ-21+uS0d+S0∑i=1nki-min⁡d,d1,…,dnu+θ-12σ12S2+∑i=1nσi+12xi2≤θ1+uθ-2∑i=1nkiS0-min⁡d,d1,…,dn+θ-12∨0σ12∨…∨σn2u2+S0d+∑i=1nkiS0-min⁡d,d1,…,dnu+d+∑i=1nkiS0=θ1+uθ-2-λu2+S0d+∑i=1nkiS0-min⁡d,d1,…,dnu+d+∑i=1nkiS0, where(22)λ=min⁡d,d1,…,dn-∑i=1nkiS0-θ-12∨0σ12∨…∨σn2>0. Let (23)A=d+∑i=1nkiS0-min⁡d,d1,…,dn be a constant. Then(24)dWu≤θ1+uθ-2-λu2+Au+d+∑i=1nkiS0dt+θ1+uθ-1σ1SdB1t+∑i=1nσi+1xidBi+1t. For 0<q<θλ, we have(25)deqtWu=LeqtWudt+eqtθ1+uθ-1σ1SdB1t+∑i=1nσi+1xidBi+1t, where (26)LeqtWu=qeqtWu+eqtLWu≤qeqt1+uθ+eqtθ1+uθ-2-λu2+Au+d+∑i=1nkiS0=eqtθ1+uθ-2-λ-qθu2+A+2qθu+S0d+∑i=1nki+qθ. Define a function f(u) by(27)fu=1+uθ-2-λ-qθu2+A+2qθu+S0d+∑i=1nki+qθ,u∈R+ and(28)F≔supu∈R+fu+1, which gives(29)LeqtWu≤θeqtF. Integrating from 0 to t and taking expectation for (25), one has(30)EeqtWu=Wu0+E∫0tLeqtWudt. Then it is easy to obtain(31)EeqtWu≤1+u0θ+E∫0teqtθFdt=1+u0θ+θFqeqt, which leads to(32)limt→∞supE1+utθ≤θFq≔F0,a.s.u(t) is a continuous function, so there is a constant M>0 such that(33)E1+utθ≤M,t≥0. According to (24), for small enough δ>0, k=1,2,…, integrating from kδ to (k+1)δ and taking expectation, we can get(34)Esupkδ≤t≤k+1δ1+utθ≤E1+ukδθ+I1+I2≤M+I1+I2, where (35)I1=Esupkδ≤t≤k+1δ∫kδtθ1+uθ-2-λu2+Au+S0d+∑i=1nkidt≤Esupkδ≤t≤k+1δ∫kδtθ1+uθ-2S0d+∑i=1nkiu+S0d+∑i=1nkidt=Esupkδ≤t≤k+1δ∫kδtθ1+uθ-2S0d+∑i=1nki1+udt≤S0d+∑i=1nkiθE∫kδk+1δ1+uθdt≤C1δEsupkδ≤t≤k+1δ1+utθ, here,(36)C1=S0d+∑i=1nkiθ. Applying Burkholder-Davis-Gundy inequality [41], we have (37)I2=Esupkδ≤t≤k+1δ∫kδtθ1+uθ-1σ1SdB1t+∑i=1nσi+1xidBi+1t≤32E∫kδk+1δθ21+u2θ-2σ12S2+∑i=1nσi+12xi2dt1/2≤32E∫kδk+1δθ21+u2θ-2σ12u2+∑i=1nσi+12u2dt1/2≤32θδ1/2Esupkδ≤t≤k+1δσ12∨…∨σn21+u2θ1/2≤32θδ1/2σ12∨…∨σn21/2Esupkδ≤t≤k+1δ1+uθ. Hence, we have(38)Esupkδ≤t≤k+1δ1+uθ≤E1+ukδθ+C1δ+32θδ1/2σ12∨…∨σn21/2×Esupkδ≤t≤k+1δ1+uθ. Particularly, one can choose δ>0 such that(39)C1δ+32θδ1/2σ12∨…∨σn21/2≤12, which yields(40)Esupkδ≤t≤k+1δ1+uθ≤2E1+ukδθ≤2M. For any β>0, the following inequality can be obtained by Chebyshev’s inequality [41], (41)Psupkδ≤t≤k+1δ1+utθ≥kδ1+β≤Esupkδ≤t≤k+1δ⁡1+utθkδ1+β≤2Mkδ1+β. Using Borel-Cantelli Lemma [41], for all ω∈Ω and sufficiently large k, we have(42)supkδ≤t≤k+1δ1+utθ≤kδ1+β. Therefore, there is a k0(ω), for all ω∈Ω and k≥k0; the above equation holds. In addition, for all ω∈Ω, when k≥k0 and kδ≤t≤(k+1)δ, (43)ln⁡1+utθln⁡t≤ln⁡kδ1+βln⁡t≤ln⁡kδ1+βln⁡kδ=1+β, which means(44)limt→∞supln⁡1+utθln⁡t≤1+β,a.s. When β→0, (45)limt→∞supln⁡1+utln⁡t≤1θ,a.s. where θ is determined by (22). Therefore, (46)limt→∞supln⁡St+∑i=1nxitln⁡t=limt→∞supln⁡utln⁡t≤limt→∞supln⁡1+utln⁡t≤1θ,a.s. This completes the proof.Lemma 4.
Ifmin⁡{d,d1,…,dn}>∑i=1nkiS0+(θ-1/2∨0)(σ12∨…∨σn2), then the solution (S(t),x1(t),…,xn(t)) of system (1) with initial value (S(0),x1(0),…,xn(0))∈R+n+1 satisfies (47)limt→∞∫0tSs-λ1dB1st=0,a.s.limt→∞∫0tx1s-x1∗dB2st=0,a.s.limt→∞∫0txisdBi+1st=0,a.s.i=2,3,…,n, where λ1 is the break-even concentration of x1(t) and x1∗ is the equilibrium concentration of x1(t).Proof.
Define(48)Xt=∫0tSs-λ1dB1s,Yt=∫0tx1s-x1∗dB2s,Zit=∫0txisdBi+1s,i=2,3,…,n. It follows from min⁡{d,d1,…,dn}>∑i=1nkiS0+(θ-1/2∨0)(σ12∨…∨σn2), Lemma 3, and Burkholder-Davis-Gundy inequality [41] that (49)Esup0≤s≤tXsθ≤CθE∫0tSs-λ12ds1/2≤Cθtθ/2Esup0≤s≤tSs-λ1θ≤Cθtθ/2Esup0≤s≤t1+utθ≤2MCθtθ/2. By Doob’s martingale inequality [41], for arbitrary α>0, it follows that (50)Pω:supkδ≤t≤k+1δXtθ>kδ1+α+θ/2≤EXk+1δθkδ1+α+θ/2≤2MCθkδ1+αk+1δkδθ/2≤21+θ/2MCθkδ1+α. By Borel-Cantelli Lemma [41], for all ω∈Ω and sufficient large k, we derive(51)supkδ≤t≤k+1δXtθ≤kδ1+α+θ/2. Thus, there exists a positive constant kX0(ω) such that for all k≥kX0 and kδ≤t≤(k+1)δ, (52)ln⁡Xtθln⁡t≤ln⁡kδ1+α+θ/2ln⁡t≤ln⁡kδ1+α+θ/2ln⁡kδ, which implies(53)ln⁡Xtθln⁡t≤1+α+θ2. Taking superior limit leads to(54)limt→∞supln⁡Xtln⁡t≤1+α+θ/2θ. Sending α→0, we have(55)limt→∞supln⁡Xtln⁡t≤12+1θ. Then there exist a constant T′(ω)>0 and a set Ωη for small enough 0<η<1/2-1/θ such that PΩη≥1-η. Thus for t≥T′(ω),ω∈Ωη, we have(56)ln⁡Xtln⁡t≤12+1θ+η, which means(57)limt→∞supXtt≤limt→∞supt1/2+1/θ+ηt=0. Together with(58)limt→∞infXtt=limt→∞inf∫0tSs-λ1dB1st≥0, we derive(59)limt→∞Xtt=0,a.s. and(60)limt→∞∫0tSs-λ1dB1st=0,a.s. Applying the same methods, we can prove that (61)limt→∞∫0tx1s-x1∗dB2st=0,a.s.limt→∞∫0txitdBi+1st=0,a.s.i=2,3,…,n. This completes the proof.Definition 5.
Stochastic equation(62)dϕt=S0-Std+∑i=1nkixidt+σ1ϕtdB1t has a solution ϕ(t) weakly converging to the distribution v. Here v is a probability measure of R+1 such that ∫0∞zv(dz)=S0. Particularly, v has density (Aσ12z2p(z))-1, where A is a normal constant and(63)pz=exp⁡2d+∑i=1nkixiS0σ12zexp⁡-2d+∑i=1nkixiS0σ12z2d+∑i=1nkixi/σ12.Theorem 6.
( S ( t ) , x 1 ( t ) , … , x n ( t ) ) is the solution of system (1) with any initial value (S(0),x1(0),…,xn(0))∈R+n+1. Moreover, if (64)0<λ1<λ2≤⋯≤λn<S0,min⁡d,d1,…,dn>∑i=1nkiS0+θ-12∨0σ12∨…∨σn2,aiλ11+biλ1+ai2bi1+biλ1Hd+N∑i=1nki1/2≤di+N∑i=1nki+σi+122i=2,3,…,n,anda1∫0∞xvdx1+b1x>σ222+d+M∑i=1nki, then we have (65)limt→∞infx1t>0,a.s.limt→∞supln⁡xitt≤0,a.s.i=2,…,n, where M=max⁡{x1,…,xn}, N=min⁡{x1,…,xn}, λi(i=1,2,…,n) is the break-even concentration of xi(t), H is determined later by (74), (λ1,x1∗,0,…,0) is the boundary equilibrium of the corresponding deterministic model, and v is defined in Definition 5.Proof.
The deterministic system corresponding to (1) has an equilibrium (λ1,x1∗,0,…,0) under the condition of λ1<S0, where λ1 and x1∗ satisfy(66)S0-λ1d+k1x1∗=a1λ1x1∗1+b1λ1,d1+k1x1∗=a1λ11+b1λ1. Define C2 function V:R+n+1→R¯+ by (67)VS,x1,…,xn=S-λ1-λln⁡Sλ1+1+b1λ1x1-x1∗-x1∗ln⁡x1x1∗+∑i=2n1+biλ1xi. Then by the Ito^ formula, we get(68)dV=S-λ1SS0-Sd+∑i=1nkixi-∑i=1naiS1+biSxidt+σ1SdB1t+12λ1S2σ12S2dt+1+b1λ1-1+b1λ1x1∗x1-d1+∑i=1nkixix1+a1S1+b1Sx1dt+σ2x1dB2t+121+b1λ1x1∗x12σ22x12dt+∑i=2n1+biλ1-di+∑i=1nkixixi+aiS1+biSxidt+σi+1xidBi+1t=LVdt+S-λ1σ1dB1t+1+b1λ1-1+b1λ1x1∗x1σ2x1dB2t+∑i=2n1+biλ1σi+1xidBi+1t, where (69)LV=S-λ1SS0-Sd+∑i=1nkixi-∑i=1naiS1+biSxi+12λ1σ12+121+b1λ1x1∗σ22+1+b1λ1-1+b1λ1x1∗x1-d1+∑i=1nkixix1+a1S1+b1Sx1+∑i=2n1+biλ1-di+∑i=1nkixixi+aiS1+biSxi. According to (66), we have (70)S0=λ1+a1λ1x1∗1+b1λ1d+k1x1∗,d1=a1λ11+b1λ1-k1x1∗. It follows from some calculations that (71)LV=-d+∑i=1nkixiS-λ12S+a1λ1x1∗S-λ1∑i=2nkixiS1+b1λ1d+k1x1∗+a1k1x1∗S-λ1x1-x1∗1+b1Sd+k1x1∗-k11+b1λ1x1-x1∗2-1+b1λ1x1-x1∗∑i=2nkixi+∑i=2nxiaiλ11+biλ1-di+∑i=1nkixi+12λ1σ12+121+b1λ1x1∗σ22≤-d+∑i=1nkixiS-λ12S+a1λ1x1∗1+b1λ1d+k1x1∗+1+b1λ1x1∗∑i=2nkixi+a1k1x1∗x1d+k1x1∗+a1k1λ1x1∗2d+k1x1∗+∑i=2nxi-∑i=1nkixi-xi∗+12λ1σ12+121+b1λ1x1∗σ22.Set(72)M=max⁡x1,…,xn,N=min⁡x1,…,xn, then (73)LV≤-d+N∑i=1nkiS-λ12S+a1λ1x1∗1+b1λ1d+k1x1∗+1+b1λ1x1∗M∑i=2nki+a1k1x1∗Md+k1x1∗+a1k1λ1x1∗2d+k1x1∗+∑i=2nM∑i=1nkixi∗+12λ1σ12+121+b1λ1x1∗σ22=-d+N∑i=1nkiS-λ12S+H, where(74)H=a1λ1x1∗1+b1λ1d+k1x1∗+1+b1λ1x1∗M∑i=2nki+a1k1x1∗Md+k1x1∗+a1k1λ1x1∗2d+k1x1∗+n-1M∑i=1nkixi∗+12λ1σ12+121+b1λ1x1∗σ22. Integrating (68) from 0 to t and dividing by t on both sides, one can obtain (75)Vt-V0t≤-d+N∑i=1nkit∫0tSs-λ12Ssds+H+σ11t∫0tSs-λ1dB1s+σ21+b1λ11t∫0tx1s-x1∗dB2t+∑i=2n1+biλ1σi+1∫0txisdBi+1t. According to Lemma 4, it follows that(76)limt→∞sup1t∫0tSs-λ12Ssds≤Hd+N∑i=1nki.
Define(77)Vixit=ln⁡xit,i=2,3,…,n. Using the Ito^ formula, we have(78)dVi=-di+∑i=1nkixi+aiS1+biS-σi+122dt+σi+1dBi+1t≤aiS-λ11+biS1+biλ1-di+N∑i=1nki+aiλ11+biλ1-σi+122dt+σi+1dBi+1t. Then integrating (78) from 0 to t on both sides and dividing all by t, we can get (79)Vit-Vi0t≤ai1+biλ1St-λ11+biSt-di+N∑i=1nki+aiλ11+biλ1-σi+122+σi+1Bi+1tt≤ai2bi1+biλ1St-λ1St-di+N∑i=1nki+aiλ11+biλ1-σi+122+σi+1Bi+1tt≤ai2bi1+biλ11t∫0tSt-λ12Stds1/2-di+N∑i=1nki+aiλ11+biλ1-σi+122+σi+1Bi+1tt, which means (80)ln⁡xitt≤ln⁡xi0t+ai2bi1+biλ11t∫0tSt-λ12Stds1/2-di+N∑i=1nki+aiλ11+biλ1-σi+122+σi+1Bi+1tt. Based on (76), limt→∞Bi+1t/t=0 and limt→∞ln⁡xi0/t=0, we know that (81)limt→∞supln⁡xitt≤-di+N∑i=1nki+aiλ11+biλ1-σi+122+ai2bi1+biλ1Hd+N∑i=1nki1/2i=2,…,n. If(82)aiλ11+biλ1+ai2bi1+biλ1Hd+N∑i=1nki1/2≤di+N∑i=1nki+σi+122, then we have(83)limt→∞supln⁡xitt≤0,a.s.i=2,…,n, that is, xi(t)(i=2,…,n) tends to extinction exponentially almost surely.
On the basis of classical comparison theorem of stochastic differential equation, the solutionS(t) of system (1) and the solution ϕ(t) of system (62) satisfy S(t)≤ϕ(t). According to the Ito^ formula, it follows that (84)dln⁡St=S0-Sd+∑i=1nkixiS-∑i=1naixi1+biS-σ122dt+σ1dB1t,dln⁡ϕt=S0-ϕd+∑i=1nϕ-σ122dt+σ1dB1t. Integrating the above two equations from 0 to t on both sides and dividing them by t, we obtain (85)ln⁡Stt=ln⁡S0t+1t∫0tS0-Sd+∑i=1nkixiS-∑i=1naixi1+biSdt-σ122+1t∫0tσ1dB1t=d+∑i=1nkixiS01S-a1x11+b1S-∑i=2naixi1+biS-d+∑i=1nkixi-σ122+σ1B1tt+ln⁡S0t,ln⁡ϕtt=ln⁡ϕ0t+1t∫0tS0-ϕd+∑i=1nkixiϕdt-σ122+σ1B1tt=d+∑i=1nkixiS01ϕ-d+∑i=1nkixi-σ122+σ1B1tt+ln⁡ϕ0t.Therefore, (86)0≥ln⁡St-ln⁡ϕtt=d+∑i=1nkixiS01S-1ϕ-a1x11+b1S-∑i=2naixi1+biS≥d+N∑i=1nkiS01S-1ϕ-a1x1-∑i=2naixi, which implies that(87)1S-1ϕ≤1d+N∑i=1nkiS0a1x1+∑i=2naixi.
Next, applying the Ito^ formula to ln⁡x1(t), one deduces (88)dln⁡x1t=-d1+∑i=1nkixi+a1S1+b1S-σ222dt+σ2dB2t≥-d1+M∑i=1nki+a1S1+b1S-σ222dt+σ2dB2t. Integrating the above inequality from 0 to t on both sides and dividing all by t, we have (89)ln⁡x1tt≥ln⁡x10t-σ222-d+M∑i=1nki+a1ϕ1+b1ϕ-a1ϕ-S1+b1ϕ1+b1S+σ2B2tt≥ln⁡x10t-σ222-d+M∑i=1nki+a1ϕ1+b1ϕ-a1b121S-1ϕ+σ2B2tt≥ln⁡x10t-σ222-d+M∑i=1nki+a1ϕ1+b1ϕ-a1b12d+N∑i=1nkiS0a1x1+∑i=2naixi+σ2B2tt≥ln⁡x10t-σ222-d+M∑i=1nki+a1ϕ1+b1ϕ-a12x1b12d+N∑i=1nkiS0+σ2B2tt. According to Lemma 3, we have(90)ln⁡x1tt≤limt→∞supln⁡x1tt≤0, which yields (91)a12x1b12d+N∑i=1nkiS0≥ln⁡x10t-σ222-d+M∑i=1nki+a1ϕ1+b1ϕ+σ2B2tt, that is, (92)limt→∞infx1t≥b12d+N∑i=1nkiS0a12-σ222-d+M∑i=1nki+a1∫0∞xvdx1+b1x. Therefore, if(93)a1∫0∞xvdx1+b1x>σ222+d+M∑i=1nki, then(94)limt→∞infx1t>0,a.s. This completes the proof.Theorem 7.
System (1) with (S(0),x1(0),…,xn(0))∈R+n+1 admits the positive solution (S(t),x1(t),…,xn(t)). Furthermore, if(95)ai∫0∞xvdx1+bix<di+N∑i=1nki+σi+122,i=1,2,…,n, then (96)limt→∞∫0tSsdst=S0,a.s.limt→∞supln⁡xitt<0,a.s.i=1,2,…,n, and S(t) weakly converges to distribution v; here v is defined as in Definition 5, which means the biomass of all species in the vessel will be extinct exponentially.Proof.
It is obvious thatS(t)≤ϕ(t) by the classical comparison theorem of stochastic differential equation, where S(t) and ϕ(t) are the solutions of system (1) and (62), separately. Applying the Ito^ formula to ln⁡xit(i=1,2,…,n), we derive that (97)dln⁡xit=-di-∑i=1nkixi+aiS1+biS-σi+122dt+σi+1dBi+1t≤-di-N∑i=1nki+aiϕ1+biϕ-σi+122dt+σi+1dBi+1t. Since aiS/1+biS is a monotonous increasing function and 0<S(t)≤ϕ(t), (98)ln⁡xitt≤-di-N∑i=1nki-σi+122+aiϕ1+biϕ+σi+1Bi+1tt+ln⁡xi0t. Using the conditions that limt→∞Bi+1t/t=0, limt→∞ln⁡xi0/t=0 and(99)ai∫0∞xvdx1+bix<di+N∑i=1nki+σi+122,i=1,2,…,n, we know that(100)limt→∞supln⁡xitt≤0,i=1,2,…,n, which means xi(t)(i=1,2,…,n) tends to zero exponentially almost surely.
Now we prove that (62) is stable in distribution. Set Y(t)=ϕ(t)-S0, then we have(101)dY=-d+∑i=1nkixiYdt+σ1S0+YdB1t. According to Theorem 2.1 in [42], we know that diffusion process Y(t) is stable in distribution when t→∞. Therefore, ϕ(t) is also stable in distribution. We further prove that S(t) is stable in distribution as t→∞. We define the following stochastic process Sε(t) with the initial value Sε(0)=S(0) by(102)dSε=S0-Sεd+∑i=1nkixi-εSεdt+σ1SεdB1t. Then(103)dS-Sε=ε-∑i=1naixi1+biSS-d+ε+∑i=1nkixiS-Sεdt+σ1S-SεdB1t. The solution of (103) is (104)St-Sεt=exp⁡σ1B1t-d+∑i=1nkixi+ε+σ122t∫0texp⁡d+∑i=1nkixi+ε+σ122s-σ1B1sε-∑i=1naixi1+biSsSsds. According to Theorem 6, when t→∞, xi→0(i=2,3,…,n), x1→x1∗>0 and S→λ1<S0. Therefore, for almost all ω∈Ω, ∃T=T(ω) such that(105)ε>∑i=1naixit1+biSt,∀t≥T. In addition, for all ω∈Ω, if t>T, it follows that (106)St-Sεt=exp⁡σ1B1t-d+∑i=1nkixi+ε+σ122t∫0Texp⁡d+∑i=1nkixi+ε+σ122s-σ1B1sε-∑i=1naixi1+biSsSsds+∫Ttexp⁡d+∑i=1nkixi+ε+σ122s-σ1B1sε-∑i=1naixi1+biSsSsds≥exp⁡σ1B1t-d+M∑i=1nki+ε+σ122t∫0Texp⁡d+N∑i=1nki+ε+σ122s-σ1B1sε-∑i=1naixi1+biSsSsds, and when t→∞, (107)exp⁡σ1B1t-d+M∑i=1nki+ε+σ122t→0,∫0Texp⁡d+N∑i=1nki+ε+σ122s-σ1B1sε-∑i=1naixi1+biSsSsds<∞, which implies that(108)limt→∞St-Sεt≥limt→∞infSt-Sεt≥0,a.s.
Now, we consider(109)dϕt-Sεt=εSεt-d+∑i=1nkixiϕt-Sεtdt+σ1ϕt-SεtdB1t≤εSεt-d+N∑i=1nkiϕt-Sεtdt+σ1ϕt-SεtdB1t. Integrating it from 0 to t, one can derive (110)ϕt-Sεt≤∫0tεSεs-d+N∑i=1nkiϕs-Sεsds+∫0tσ1ϕs-SεsdB1s. Taking expectation for both sides of the above inequality, we obtain (111)Eϕt-Sεt≤E∫0tεSεs-d+N∑i=1nkiϕs-Sεsds≤E∫0tεϕs-d+N∑i=1nkiϕs-Sεsds, then(112)Eϕt-Sεt≤εsup0≤sEϕs, which leads to(113)limε→0inflimt→∞Eϕt-Sεt=0. According to (108), (113), and S(t)≤ϕ(t), we can get(114)limt→∞ϕt-St=0. Therefore, based on the fact that ϕ(t) is stable to distribution v as t→∞, S(t) weakly converges to distribution v. This completes the proof.


## 4. 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 xi(i=2,…,n) in the vessel will be extinct and x1 will survive due to the fact that xi(i=2,…,n) is inadequate competitor [2] if 0<λ1<λ2≤⋯≤λn<S0.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 populationxi(t)(i=1,2,…,n) in system (1) survives and the other microorganism populations in the vessel will be extinct if the following general condition holds:(115)0<λi<λj≤⋯≤λk<S0,i,j,k=1,…,nandi≠j≠k.For a better understanding to the results obtained in Theorem6, we offer an example as follows.LetS0=1.2, d=0.12, d1=0.32, d2=0.31, a1=2, b1=3, a2=2, b2=5, σ1=0.1, σ2=0.1, σ3=0.2, k1=0.007, and k2=0.01, then system (1) becomes(116)dSt=1.2-St0.12+0.007x1t+0.01x2t-2St1+3Stx1t-2St1+5Stx2tdt+0.1StdB1t,dx1t=-0.32+0.007x1t+0.01x2tx1t+2St1+3Stx1tdt+0.1x1tdB2t,dx2t=-0.31+0.007x1t+0.01x2tx2t+2St1+5Stx2tdt+0.2x2tdB3t.We derive x1∗=0.1913, x2∗=0, λ1=0.3102, and λ2=0.7128. Obviously, λ1<λ2<S0. The concentration of nutrient at equilibrium is S∗=λ1=0.3102. Hence, the species besides x1(t) will be extinct eventually as Theorem 6 shows (see Figure 1). From the biological viewpoint, the natural resource for population will be fluctuated because of all kinds of reasons such as excessive development or natural disasters. This phenomenon also occurs in microorganisms cultivation. If the temperature or lightness is various (the density of white noise σi represents this phenomenon in system (1)), the dynamical behaviors, such as digestion ability for microorganisms, will be various at different points. That is, the stochastic factors will cause the variations of dynamical behaviors and this phenomenon is interacted between nutrient S(t) and microorganisms xi(t) in the turbidostat. The stochastic factors always cause the fluctuation of S(t) and xi(t) (see Figures 1–5).Figure 1
x 1 survive and x2 will be extinct for system (116) with initial value S0=0.5, x1(0)=x2(0)=0.3. (a) and (b) are the time series of system with different time scale.
(a) (b)Figure 2
(a) is the time series ofS(t) with initial value S(0)=0.5; (b) is the distribution interval of S(t).
(a) (b)Figure 3
(a) is the time series ofS(t) with initial value S(0)=0.5; (b) is the distribution interval of S(t).
(a) (b)Figure 4
(a) is the time series ofx1(t) with initial value x1(0)=0.3; (b) is the distribution interval of x1(t).
(a) (b)Figure 5
(a) is the time series ofx1(t) with initial value x1(0)=0.3; (b) is the distribution interval of x1(t).
(a) (b)Although the concentration of nutrients varies significantly in the system, it always fluctuates aroundλ1=0.3102 (the black line in Figures 1(b), 2(a), and 3(a)) and has a stationary distribution in the end (see Figures 2(b) and 3(b)). The concentration of x1(t) fluctuates around x1∗=0.1913 (the black line in Figure 1(b)), and there exists a stationary distribution (see Figures 4(b) and 5(b)). This fluctuation phenomenon basically comes from the random factors which leads to variation of concentration for S(t) and xi(t). The species x2(t) will be extinct since the concentration of nutrient S∗=λ1=0.3102 is less than its break-even concentration λ2=0.7128 (see Figure 1).To further confirm the results obtained in Theorem7, we give the following example.SetS0=1.2, d=0.8, d1=0.9, d2=0.9, a1=1, b1=0.4, a2=1, b2=0.6, σ1=0.2, σ2=0.8, σ3=0.8, k1=0.2, and k2=0.2, then system (1) arrives at(117)dSt=1.2-St0.8+0.2x1t+0.2x2t-St1+0.4Stx1t-St1+0.6Stx2tdt+0.2StdB1t,dx1t=-0.9+0.2x1t+0.2x2tx1t+St1+0.4Stx1tdt+0.8x1tdB2t,dx2t=-0.9+0.2x1t+0.2x2tx2t+St1+0.6Stx2tdt+0.8x2tdB3t.It follows from simple calculations to obtain 1/ai[di+N(k1+k2)+σi+12/2]≈1.22>S0, which satisfies the conditions of Theorem 7. Hence, x1(t) and x2(t) tend to zero exponentially, which implies x1∗=0, x2∗=0 (see Figure 6).Figure 6
x 1 and x2 die out for system (117) with initial value S(0)=0.5, x1(0)=x2(0)=0.3. (a) and (b) are the time series of system with different time scale.
(a) (b)In fact, if the density of white noiseσi+1 increased (such as extremely high temperature) so that 1/ai[di+N(k1+k2)+σi+12/2]>S0; 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.



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