# Stationary Distribution and Extinction of a Stochastic Microbial Flocculation Model with Regime Switching

Authors: Haisu Zhang; Yi Song
Journal: Complexity
 (2021)

---

## Abstract

In this paper, a stochastic microbial flocculation model with regime switching is developed and analyzed. By proposing a suitable stochastic Lyapunov function, the existence and ergodicity of a stationary distribution for the system are proved. Then, the extinction of microorganisms is discussed under appropriate conditions and sufficient conditions for extinction are obtained. Finally, the results of the theoretical analysis are illustrated by numerical simulation.


---

## Body

## 1. Introduction

The flocculation process is a phenomenon in which small clumps usually aggregate into large clumps in a liquid medium [1]. Usually, a certain substance is added to the solid suspension, which can promote the occurrence of flocculation, and is generally called a flocculant. Flocculants have important applications in sewage treatment, energy extraction, biopharmacy, aquaculture, etc. [2, 3, 4]. For example, Gupta and Ako explored the effect of guar gum flocculant in the treatment of drinking water or food processing water through experiments and found that guar gum flocculant can not only improve the quality of water in the process of drinking water clarification, and there is no residue of acrylamide in the water, which reduces the health risks of the population [5]. Wu et al. used chitosan flocculant to treat the Mn(II) and suspended solids produced in the dual-alkali flue gas desulfurization regeneration process, which solved the problem that the traditional methods (such as chemical precipitation method, ion exchange method, adsorption method, and electrodeposition method) are difficult to remove heavy metal ions in the suspension [6]. Based on the flocculation precipitation method, Dharani and Balasubramanian synthesized a new water-soluble flocculant for harvesting microalgae, which is inexpensive, environmentally friendly, easy to synthesize, and has high harvesting efficiency for microalgae, compared with other flocculants [7].The above research is generally based on experiments to study the flocculation effect of different flocculants or develop new flocculants. Recently, some scholars have begun to pay attention to the complex dynamic behavior in the process of microbial flocculation mathematically [8–17]. On the basis of the chemostat model [18–21], Tai et al. [9] proposed the following model:(1)dStdt=DS0−St−μ1StXt,dXtdt=μ2St−τXt−τ−DXt−h2XtPt,dPtdt=DP0−Pt−h3XtPt,to describe the process of microbial flocculation, where S, X, and P represent the concentration of nutrition, microorganisms, and flocculants, respectively. For the meaning of all parameters, refer to Tai et al. [9]. Based on system (1) and motivated by the authors in [22, 23], Zhang et al. [14, 15] considered the following models from a random view:(2)dSt=DS0−St−μ1StXtdt+σ1StdB1t,dXt=μ2StXt−DXt−h2XtPtdt+σ2XtdB2t,dPt=DP0−Pt−h3XtPtdt+σ3PtdB3t,(3)dSt=DS0−St−μ1StXtdt+σ1StdB1t,dXt=μ2e−DτSt−τXt−τ−DXt−h2XtPtdt+σ2XtdB2t,dPt=DP0−Pt−h3XtPtdt+σ3PtdB3t,where Bit are independent standard Brownian motions and Bi0=0i=1,2,3 and σ1>0,σ2>0 and σ3>0 are the intensities of the white noise on the nutrition, microorganisms, and flocculants, respectively. For system (2), the authors discussed the existence and ergodicity of a stationary distribution, and for system (3), the authors investigated the asymptotic behaviors of the solutions.However, there is another kind of noise interference in nature, namely, telegraph noise or colored noise, which is used to express the transformation of system variables from one state to another. The development of microorganisms in a thermostat is not immune to the effects of temperature, humidity, or light. At the microscopic stage, the system constantly transitions from one state to another. Therefore, many researchers have focused on the chemostat model with telegraph noise and achieved good results [24–26].In this paper, by using the method in Wang and Jiang [27], by considering the Markov regimes in the velocity D in system (3), we get the new system as follows:(4)dSt=DrtS0−St−μ1StXtdt+σ1rtStdB1t,dXt=μ2StXt−DrtXt−h2XtPtdt+σ2rtXtdB2t,dPt=DrtP0−Pt−h3XtPtdt+σ3rtPtdB3t,where rt,t≥0 represent the continuous-time Markov chain independent from the Brownian motion Bt and takes a value in finite-state space S=1,2,…,n. Dk>0 and σik>0,i=1,2,3 hold for all k∈S.


## 2. Preliminary

For a right-continuous Markov chainrt, the generator Υ=γijn×n is determined by(5)Prt+Δt=j|rt=i=γijΔ+oΔ,ifi≠j,1+γijΔ+oΔ,ifi=j,where Δt>0,γij≥0 is the transition rate from i to j if i≠j while γii=−∑i≠jγij. In order to ensure that rt is irreducible, here, we need to assume γij≥0, for i≠j. Thus, rt has a unique stationary distribution π=π1,π2,…,πn with the form(6)πΥ=0,subject to(7)∑h=1nπh=1πh>0,∀h∈S.For a diffusion processut,rt expressed by stochastic differential equations,(8)dut=gut,rtdt+hut,rtdBt,u0=u0,r0=r,where g⋅,⋅:ℝn×S⟶ℝn,g⋅,⋅:ℝn×S⟶ℝn×d satisfying hu,khTu,k=piju,k. For each k∈S, let V⋅,k be any twice continuously differentiable function and we can define the operator L as follows:(9)LVu,k=∑i=1ngiu,k∂Vu,k∂ui+12∑i,j=1npiju,k∂2Vu,k∂ui∂zj+∑l=1NγklVu,l.To simplify, we definegt=1/t∫0tgsds,t>0, where gt is an integral function define on 0,∞. And we define gl=inft∈0,∞gt and gu=supt∈0,∞gt, where gt is a bounded function on 0,∞. We denote h^=mink∈Shk and hˇ=maxk∈Shk, where hkk∈S is a constant vector. Then according to the ergodicity of Markov chain r⋅, one gets(10)limt⟶∞1t∫0thrsds=∑k∈Sπkhka.s.


## 3. Existence and Uniqueness of Positive Solution

Theorem 1.
For any given initial valueS0,X0,P0,r0∈ℝ+3×S, there is a unique positive solution St,Xt,Pt,rt of system (4) on t≥0 and the solution will remain in ℝ+3×S with probability one.Proof.
Firstly, letℏe be the explosion time, we claim that there exists a unique local solution St,Xt,Pt,rt∈ℝ+3×S with initial value S0,X0,P0,r0∈ℝ+3×S on t∈0,ℏe a.s. In fact, it is easy to get from the local Lipschitz property of the coefficients of system (4).
Secondly, we proveℏe=∞ a.s.; it means the solution is global. Let n0≥1 be sufficiently large such that S0,X0,P0,r0 all lie within the interval 1/n0,n0 For each integer n≥n0, define the stopping time(11)ℏn=inft∈0,ℏe:St∉1n,norXt∉1n,norPt∉1n,n.
In the full text, we denoteinf∅=∞ (the empty set). Clearly, ℏn is increasing as n⟶∞. Let ℏ∞=limn⟶∞ℏn, whence ℏ∞≤ℏe a.s. If ℏ∞=∞ a.s. is true, then ℏe=∞ and St,Xt,Pt,rt∈ℝ+3×S a.s. for all t≥0. Then, to achieve our purpose, we only need to show ℏ∞=∞ a.s. If this affirmation is not true, then there exist two constants T>0 and ϵ∈0,1 such that(12)ℙℏ∞≤T>ϵ.
Thus, for an integern1≥n0, we have(13)ℙℏn≤T>ϵ,∀n≥n1.
Define aC2-function V:ℝ+3⟶ℝ+ as follows:(14)VS,X,P=S−a1−a1lnSa1+a2X−a3−a3lnXa3+P−a4−a4lnPa4,where a1=2D^/3μ2,a2=μ1/μ2,a3=μ2D^/μ1h2,a4=μ1D^/3μ2h3. From the Itô’s formula, we have(15)dVS,X,P=ℒVS,X,Pdt+1−2D^3μ2Sσ1rtSdB1t+μ1μ2−D^h2Xσ2rtXdB2t+1−μ1D^3μ2h3Pσ3rtPdB3t,where ℒV:ℝ+3⟶ℝ+ is defined by(16)ℒV=1−2D^3μ2SDkS0−DkS−μ1SX+μ1μ2−D^h2Xμ2SX−DkX−h2XP+1−μ1D^3μ2h3PDkP0−DkP−h3XP+D^σ12k3μ2+D^σ22k2h2+μ1D^σ32k6μ2h3=DkS0−DkS−μ1SX−2D^DkS03μ2S+2D^Dk3μ2+2μ1D^3μ2X+D^σ12k3μ2+μ1SX−μ1Dkμ2X−μ1h2μ2XP−μ2D^h2S+D^Dkh2+D^P+D^σ22k2h2+DkP0−DkP−h3XP−μ1D^DkP03μ2h3P+μ1D^Dk3μ2h3+μ1D^3μ2X+μ1D^σ32k6μ2h3≤DˇS0+2D^Dˇ3μ2+D^Dˇh2+DˇP0+μ1D^Dˇ3μ2h3+D^σˇ123μ2+D^σˇ222h2+μ1D^σˇ326μ2h3+D^P−DkP+μ1D^μ2X−μ1Dkμ2X≤DˇS0+2D^Dˇ3μ2+D^Dˇh2+DˇP0+μ1D^Dˇ3μ2h3+D^σˇ123μ2+D^σˇ222h2+μ1D^σˇ326μ2h3≔K1.
Integrating (15) from 0 to Tmin=ℏn∧T=minℏn,T and taking the mathematical expectation on both sides, we have(17)EVSTmin,XTmin,PTmin≤VS0,X0,P0+K1ETmin.
Hence,(18)EVSTmin,XTmin,PTmin≤VS0,X0,P0+K1T.
LetΩn=ω∈Ω:ℏn=ℏnω≤T for n≥n1 and according to (13), we get ℙΩn≥ϵ. Note that, for every ω∈Ωn, there exist Sℏn,ω or Xℏn,ω or Pℏn,ω equals either 1/n or n. Therefore, VSℏn,ω,Xℏn,ω,Pℏn,ω is no less than either(19)1n−1−ln1n,orn−1−lnn.
Therefore, we have(20)VSℏn,ω,Xℏn,ω,Pℏn,ω≥1n−1+lnn∧n−1−lnn.
By (18), we obtain(21)VS0,X0,P0+K1T≥E1ΩnωVSτn,ω,Xτn,ω,Pτn,ω≥ε1n−1+lnn∧n−1−lnn,where 1Ωn is the indicator function of Ωn. Let n⟶∞, we obtain(22)∞>VS0,X0,P0+K1T=∞,a contradiction. Thus, we obtain τ∞=∞ a.s. This completes the proof.


## 4. Existence of an Ergodic Stationary Distribution

In this section, we explore the existence and ergodicity of a stationary distribution for system (4). Firstly, by Lemma 3.2 in [28], we need to construct a new system, which has the same ergodic property and positive recurrence as the original system. To this end, we make a transformation αt=logSt,βt=logXt,ζt=logPt, then system (4) can be change to(23)dαt=DrtS0e−αt−Drt−μ1eβt−12σ12rtdt+σ1rtdB1t,dβt=μ2eαt−Drt−h2eζt−12σ22rtdt+σ2rtdB2t,dζt=DrtP0e−ζt−Drt−h3eβt−12σ32rtdt+σ3rtdB3t.Denote(24)Λk=DkS0μ2Dk+σ12k/2−Dk−h2P0−σ22k2,and we have the following theorem.Theorem 2.
IfΛ=∑k∈SπkΛk>0, then the stochastic process St,Xt,Pt,rt of system (3) with any initial value S0,X0,P0,r0∈ℝ+3×S is ergodic and produce a unique stationary distribution in ℝ+3×S.Proof.
The proof is divided into three steps. We have to check whether the conditions of Lemma 2.1 in [28] are satisfied one by one. In the first step, by hypothesis γij>0 for i≠j, we can see condition (i) in Lemma 2.1 in [28] is met. Secondly, choose gz,k=digσ1k,σ2k,σ3kk∈S, and then gz,kgTz,k=digσ12k,σ22k,σ32k is positive definite; obviously, condition (ii) in Lemma 2.1 in [28] is met. At the last step, we will show condition (iii) in Lemma 2.1 in [28] is true.
Let us choose a constantξ∈0,1 sufficiently small such that(25)AD^−ξ+12σˇ12∨σˇ22∨σˇ32>0,and M>0 sufficiently big such that(26)BΦu−MΛ≤−2.
Now we construct a continuous function(27)V˜α,β,ζ,k=MV1+V2+V3+V4,where(28)V1=−a1α−β+h2Dkeζ+a1μ1D^eβ,V2=ϖk+ϖ,V3=1ξ+2μ2eα+μ1eβ+eζξ+2,V4=−ζ,a1=DkS0μ2Dk+σ12k/22.
SinceV˜α˜,β˜,ζ˜,k is the minimum value point of function V˜α,β,ζ,k, then, V˜α,β,ζ,k−V˜α˜,β˜,ζ˜,k≥0. We construct a C2-function V:ℝ+3×S⟶ℝ+ with(29)Vα,β,ζ,k=V˜α,β,ζ,k−V˜α˜,β˜,ζ˜,k+Mϖk+ϖ,where ϖ=ϖ1,ϖ2,…,ϖnT, ϖ=ϖ12+ϖ22+⋯+ϖn2, and ϖkk∈S will be determined as follows. ϖ here is used to make sure that ϖk+ϖ is nonnegative. Therefore, the C2-function Vα,β,ζ,k is nonnegative.
By using the Itô’s formula, we have(30)LVα,β,ζ,k=MLV1+V2+LV3+LV4.
From system (23) and by using Itô’s formula, we have(31)L−a1α−β+h2Dkeζ=−a1DkS0e−α+a1Dk+a1μ1eβ+a12σ12k−μ2eα+Dk+h2eζ+12σ22k+h2P0−h2eζ−h2h3eβeζDk≤−a1DkS0e−α−μ2eα+a1μ1eβ+a1Dk+a12σ12k+h2P0+Dk+12σ22k≤−2a1DkS0μ2+a1Dk+σ12k2+Dk+h2P0+σ22k2+a1μ1eβ≤−DkS0μ2Dk+σ12k/2−Dk−h2P0−σ22k2+a1μ1eβ≔−Λk+a1μ1eβ.
In view of the second equation on (23), we have(32)Leβ=μ2eαeβ−Dkeβ−h2eζeβ≤μ2eαeβ−D^eβ.
Therefore, we get(33)LV1≤−Λk+a1μ1μ2D^eαeβ.
Similarly, according to the Itô’s formula, we have(34)LV2=∑l≠k∈Sγklϖl−ϖk.
Next, let us define the vectorΩ=Ω1,Ω2,…,Ωn with elements Ωk=−Λ1k,k=1,2,…,n. Since the generator matrix Υ is irreducible, then for each Λk, there exists one solution ϖ=ϖ1,ϖ2,…,ϖnT of the Poisson system (see Lemma 2.3 [29]) such that(35)Υϖ+Ω=∑j=1nπjΩj1⟶,where 1⟶ is the unit vector. Taking advantage of πΥ=0, one gets(36)−Λk+∑l≠k∈Sγklϖl−ϖk=−∑k∈SπkΛk=−Λ.
Then, we obtain that(37)LV1+V2≤−Λ+a1μ1μ2D^eαeβ.
In the meanwhile, denoteQα,β,ζ=μ2eα+μ1eβ+eζ, and we have(38)LV3≤Qξ+1α,β,ζDkμ2S0−Dkμ2eα−Dkμ1eβ+DkP0−Dkeζ+ξ+12σ12kμ2eα2Qξα,β,ζ+ξ+12σ22kμ1eβ2Qξα,β,ζ+ξ+12σ32keζ2Qξα,β,ζ≤Dˇμ2S0+DˇP0Qξ+1α,β,ζ−D^Qξ+2α,β,ζ+ξ+12Qξα,β,ζσˇ12∨σˇ22∨σˇ32Q2α,β,ζ=Dˇμ2S0+DˇP0Qξ+1α,β,ζ−D^−ξ+12σˇ12∨σˇ12∨σˇ32Qξ+2α,β,ζ.
By using the Itô’s formula toV4, we obtain(39)LV4=−DkP0eζ+Dk+h3eβ+12σ32k≤−D^P0eζ+Dˇ+h3eβ+12σˇ32.
According (30) and (37)–(39), we have(40)LV≤Φα,β,ζ+Ψα,β,where(41)Φα,β,ζ=Dˇμ2S0+DˇP0μ2eα+μ1eβ+eζξ+1−D^P0eζ+Dˇ+h3eβ+12σˇ32−D^−ξ+12σˇ12∨σˇ12∨σˇ32μ2eα+μ1eβ+eζξ+2,Ψα,β=M−Λ+a1μ1μ2D^eαeβ.
According to conditionA, we derive that(42)limζ⟶+∞Φα,β,ζ+Ψu=−∞,limβ⟶+∞Φα,β,ζ+limβ⟶+∞Ψα,β=−∞,limα⟶+∞Φα,β,ζ+limα⟶+∞Ψα,β=−∞,limζ⟶−∞Φα,β,ζ+Ψu=−∞.
In the same way, we get(43)limβ⟶−∞Φα,β,ζ+limβ⟶−∞Ψα,β=Φu−MΛ≤−2,limα⟶−∞Φα,β,ζ+limα⟶−∞Ψα,β=Φu−MΛ≤−2.
Let us consider a compact subsetU⊂ℝ+3 as follows:(44)U=logϵ<α<log1ϵ,logϵ<β<log1ϵ,logϵ<ζ<log1ϵ,where ϵ is small enough. Then we obtain(45)LVα,β,ζ,k<−1,α,β,ζ,k∈ℝ+3\U×S.
In summary, the conditions of Lemma 2.1 in [28] are fully satisfied, and then we conclude that diffusion process αt,βt,ζt,rt for system (23) is ergodic. Hence, stochastic process St,Xt,Pt,rt produces an ergodic stationary distribution. The proof is completed.


## 5. Extinction

Lemma 1 (see [30]).
LetN=Ntt≥0 be a real-valued continuous local martingale vanishing at t=0. Then,(46)limt⟶∞N,Nt=∞a.s.⇒limt⟶∞NtN,Nt=0a.s.,limt⟶∞N,Ntt<∞a.s.⇒limt⟶∞Ntt=0a.s.The following lemma is essentially the same as that in [31], so we omit it.Lemma 2.
LetSt,Xt,Pt,rt be the solution of system (3) with any initial value S0,X0,P0,r0∈ℝ+3×S if the following hypothesis K1 holds.
K1: there is a constant ι>2 such that(47)D^−12ι−1σˇ12∨σˇ22∨σˇ32>0,then the solution St,Xt,Pt,rt has the following properties:(48)limt⟶∞Stt=0,limt⟶∞Xtt=0,limt⟶∞Ptt=0,limt⟶∞1t∫0tσ1rsSsdB1s=0,limt⟶∞1t∫0tσ1rsXsdB2s=0,limt⟶∞1t∫0tσ1rsPsdB3s=0,a.s.Theorem 3.
Let hypothesisK1 hold. If μ2S0Dˇ/D^+P0Dˇ/D^<∑k∈SπkDk+1/2σ22k, then the solution St,Xt,Pt,rt for stochastic system (4) with any initial value S0,X0,P0,r0∈ℝ+3×S, will have the property that(49)limt⟶∞Xt=0,a.s., which implies the microorganism Xt tends to extinction exponentially with probability one.Proof.
Respectively, from each equation of system (4), we can obtain(50)μ2St−S0t=μ2S0Drt−μ2DrtSt−μ1μ2StXt+μ2t∫0tσ1rsSsdB1s,μ1Xt−X0t=μ1μ2StXt−μ1DrtXt−μ1h2XtPt+μ1t∫0tσ1rsXsdB2s,Pt−P0t=P0Drt−DrtPt−h3XtPt+1t∫0tσ1rsPsdB3s.
Add the left side of the three equations above to get(51)μ2St−S0t+μ1Xt−X0t+Pt−P0t≤μ2S0Drt−μ2DrtSt−μ1DrtXt+P0Drt−DrtPt+ω1+ω2+ω3,where(52)ω1=μ2t∫0tσ1rsSsdB1s,ω2=μ1t∫0tσ1rsXsdB2s,ω3=1t∫0tσ1rsPsdB3s,then(53)μ2DrtSt+μ1DrtXt+DrtPt≤μ2S0Drt+P0Drt−φ1t,
In the sequel,(54)μ2St+μ1Xt+Pt≤μ1S0DˇD^+P0DˇD^−φ1tD^,where(55)φ1t=μ2St−S0t+μ1Xt−X0t+Pt−P0t−ω1−ω2−ω3.
According to Lemma2, we have(56)limt⟶∞φ1t=0a.s.
By using the Itô’s formula and utilizing (54), we derive that(57)lnXt=lnX0+μ2Stt−Drt+12σ22rtt−h2Ptt+∫0tσ2rsdB2s≤lnX0−μ1Xtt−Ptt+μ1S0DˇD^t+P0DˇD^t−φ1tD^t−Drt+12σ22rtt+∫0tσ2rsdB2s≤lnX0+μ1S0DˇD^t+P0DˇD^t−φ1tD^t−Drt+12σ22rtt+∫0tσ2rsdB2s.
Let us divide byt on both sides of the above formula, and we can get(58)lnXtt≤lnX0t+μ1S0DˇD^+P0DˇD^−φ1tD^−Drt+12σ22rt+M2tt,where Mit=∫0tσirsdBisi=1,2,3 satisfies(59)Mi,Mi=∫0tσi2rsds<σiˇ2t.
Together with Lemma1, we can get(60)limt⟶∞Mitt=0a.s.
On the other hand, from (10), we have(61)limt⟶∞1t∫0tDrs+12σ22rsds=∑k∈SπkDk+12σ22k.
Take the limit on both sides of (58); if μ2S0Dˇ/D^+P0Dˇ/D^<∑k∈SπkDk+1/2σ22k, then we have(62)limsupt⟶∞lnXtt≤μ2S0DˇD^+P0DˇD^−∑k∈SπkDk+12σ22k<0a.s.
Hence, we get(63)limt⟶∞Xt=0a.s.
This completes the proof.


## 6. Numerical Simulations

Now, we use numerical simulations to verify the results previously obtained. A discretization for system (3) implies(64)Sk+1=Sk+DriΔtS0−Sk−μ1SkXkΔt+σ1riΔtSkΔB1,k,Xk+1=Xk+μ2SkXk−DriΔtXk−h2XkPkΔt+σ2riΔtXkΔB2,k,Pk+1=Pk+DriΔtP0−Pk−h3XkPkΔt+σ3riΔtPkΔB3,k,where ΔBi,k≜Btk+1−Btki=1,2,3 obeys the Gaussian distribution N0,Δt.Firstly, we consider the cases of two regimes, without loss of generality, we letk=1 and k=2, and the switching between them is governed by rt in the state space S=1,2 with the generator(65)−664−4.A direct computation shows the stationary distribution isπ=π1,π2=0.4,0.6.Example 1.
For system (3), the parameters are chosen as(66)S0=3.2,P0=1.2,μ1=1.6,μ2=1.2,h2=2,h3=0.5,D1=0.5,σ11=σ21=σ31=0.05,D2=0.6,σ12=σ22=σ32=0.05,with the initial values S0,X0,P0=0.8,0.2,0.3.
A direct computation showsΛ=∑k∈SπkΛk=0.8702>0, and then, by Theorem 2, system (4) has a unique ergodic stationary distribution (see Figure 1). Compared the solutions of the deterministic system, it can be seen that the stochastic disturbance has a significant impact on system (4) (see Figure 2).Figure 1
Numerical simulation of system (4). (a) Markov chain, (b) the density function of S, (c) the density function of X, and (d) the density function of P.
(a)(b)(c)(d)Figure 2
The comparison of solutions for system (1) and system (4). (a) The solutions of system (1) with τ=0, (b) the solutions of system (4), and (c) the comparison of phase portrait for system (1) and system (4).
(a)(b)(c)Example 2.
For system (4), choose the parameters as(67)S0=0.8,P0=0.6,μ1=1.8,μ2=1.4,h2=2,h3=0.5,D1=2,σ11=0.15,σ21=0.2,σ31=0.1,D2=2.4,σ12=0.2,σ22=0.24,σ32=0.1,and the initial values are same as Example 1.
By simple computation, we getμ2S0Dˇ/D^+P0Dˇ/D^=2.064<∑k∈SπkDk+1/2σ22k=2.26528, and according to Theorem 3, the microorganism will eventually become extinct (see Figure 3).Figure 3
The numerical simulation of solutions of system (4). (a) Markov chain, (b) the solution of system (4), (c) the density function of S(t) in system (4), (d) the density function of X(t) in system (4), and (e) the density function of Pt in system (4).
(a)(b)(c)(d)(e)


## 7. Conclusion

In this paper, focusing on the dynamics of the microbial flocculation process, a new type of chemostat model with flocculation effect is proposed from a random perspective. The stochastic dynamic properties of the system are investigated under regime switching conditions. The existence and ergodicity for a stationary distribution of the system and the extinction of the microorganisms are proved, and the corresponding sufficient conditions are obtained. Moreover, the theoretical results are illustrated by computer simulation. Our results enrich the research work of microbial culture and flocculation problems. However, due to the complexity of the model, the durability of the system has not been proven, and we leave this issue for future research.



---

*Source: 4920018-2021-11-22.xml*