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

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.


Introduction
e 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]. e 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][9][10][11][12][13][14][15][16][17]. On the basis of the chemostat model [18][19][20][21], Tai et al. [9] proposed the following model: 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: where B i (t) are independent standard Brownian motions and B i (0) � 0 (i � 1, 2, 3) and σ 1 > 0, σ 2 > 0 and σ 3 > 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. e 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. erefore, many researchers have focused on the chemostat model with telegraph noise and achieved good results [24][25][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: where r(t), t ≥ 0 represent the continuous-time Markov chain independent from the Brownian motion B(t) and takes a value in finite-state space S � 1, 2, . . . , n { }. D(k) > 0 and σ i (k) > 0, i � 1, 2, 3 hold for all k ∈ S.

Preliminary
For a right-continuous Markov chain r(t), the generator Υ � (c ij ) n×n is determined by where In order to ensure that r(t) is irreducible, here, we need to assume c ij ≥ 0, for i ≠ j. us, r(t) has a unique stationary distribution π � π 1 , π 2 , . . . , π n with the form subject to n h�1 π h � 1 π h > 0, ∀h ∈ S.
Proof. Firstly, let Z e be the explosion time, we claim that there exists a unique local solution s. In fact, it is easy to get from the local Lipschitz property of the coefficients of system (4).
s. for all t ≥ 0. en, to achieve our purpose, we only need to show Z ∞ � ∞ a.s. If this affirmation is not true, then there exist two constants T > 0 and ϵ ∈ (0, 1) such that us, for an integer n 1 ≥ n 0 , we have Define a C 2 -function V: R 3 + ⟶ R + as follows: where where LV: Integrating (15) from 0 to T min � Z n ∧T � min Z n , T and taking the mathematical expectation on both sides, we have Hence, Let Ω n � ω ∈ Ω: Z n � Z n (ω) ≤ T for n ≥ n 1 and according to (13), we get P(Ω n ) ≥ ϵ. Note that, for every ω ∈ Ω n , there exist S(Z n , ω) or X(Z n , ω) or P(Z n , ω) equals either 1/n or n. erefore, V(S(Z n , ω), X(Z n , ω), P(Z n , ω)) is no less than either erefore, we have By (18), we obtain 4 Complexity where 1 Ω n is the indicator function of Ω n . Let n ⟶ ∞, we obtain a contradiction. us, we obtain τ ∞ � ∞ a.s. is completes the proof.

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 and we have the following theorem.

Extinction
Lemma 1 (see [30]). Let N � N t t ≥ 0 be a real-valued continuous local martingale vanishing at t � 0. en, e following lemma is essentially the same as that in [31], so we omit it.
Proof. Respectively, from each equation of system (4), we can obtain Add the left side of the three equations above to get where then μ 2 〈D(r(t))S(t)〉 + μ 1 〈D(r(t))X(t)〉 +〈D(r(t))P(t)〉 ≤ μ 2 S 0 〈D(r(t))〉 + P(0)〈D(r(t))〉 − φ 1 (t), In the sequel, 8 Complexity where According to Lemma 2, we have By using the Itô's formula and utilizing (54), we derive that Let us divide by t on both sides of the above formula, and we can get where M i (t) � t 0 σ i (r(s))dB i (s) (i � 1, 2, 3) satisfies Together with Lemma 1, we can get On the other hand, from (10), we have Take the limit on both sides of (58); if

Complexity
Hence, we get is completes the proof.

Numerical Simulations
Now, we use numerical simulations to verify the results previously obtained. A discretization for system (3) implies 10 Complexity Firstly, we consider the cases of two regimes, without loss of generality, we let k � 1 and k � 2, and the switching between them is governed by r(t) in the state space S � (1, 2) with the generator A direct computation shows the stationary distribution is π � (π 1 , π 2 ) � (0.4, 0.6). (3), the parameters are chosen as

Complexity
A direct computation shows Λ � k∈S π k Λ (k) � 0.8702 > 0, and then, by eorem 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). Example 2. For system (4), choose the parameters as S 0 � 0.8, P 0 � 0.6, μ 1 � 1.8, and the initial values are same as Example 1.

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. e stochastic dynamic properties of the system are investigated under regime switching conditions. e 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.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this study.