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.
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:to describe the process of microbial flocculation, where , , and 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 are independent standard Brownian motions and and and 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 in system (3), we get the new system as follows:where represent the continuous-time Markov chain independent from the Brownian motion and takes a value in finite-state space . and hold for all .
2. Preliminary
For a right-continuous Markov chain , the generator is determined bywhere is the transition rate from to if while . In order to ensure that is irreducible, here, we need to assume , for . Thus, has a unique stationary distribution with the formsubject to
For a diffusion process expressed by stochastic differential equations,where satisfying . For each , let be any twice continuously differentiable function and we can define the operator as follows:
To simplify, we define , where is an integral function define on . And we define and , where is a bounded function on . We denote and , where is a constant vector. Then according to the ergodicity of Markov chain , one gets
3. Existence and Uniqueness of Positive Solution
Theorem 1. For any given initial value , there is a unique positive solution of system (4) on and the solution will remain in with probability one.
Proof. Firstly, let be the explosion time, we claim that there exists a unique local solution with initial value on a.s. In fact, it is easy to get from the local Lipschitz property of the coefficients of system (4).
Secondly, we prove a.s.; it means the solution is global. Let be sufficiently large such that all lie within the interval For each integer , define the stopping timeIn the full text, we denote (the empty set). Clearly, is increasing as . Let , whence a.s. If a.s. is true, then and a.s. for all . Then, to achieve our purpose, we only need to show a.s. If this affirmation is not true, then there exist two constants and such thatThus, for an integer , we haveDefine a -function as follows:where . From the Itô’s formula, we havewhere is defined byIntegrating (15) from 0 to and taking the mathematical expectation on both sides, we haveHence,Let for and according to (13), we get . Note that, for every , there exist or or equals either or . Therefore, is no less than eitherTherefore, we haveBy (18), we obtainwhere is the indicator function of . Let , we obtaina contradiction. Thus, we obtain 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 , then system (4) can be change to
Denoteand we have the following theorem.
Theorem 2. If , then the stochastic process of system (3) with any initial value is ergodic and produce a unique stationary distribution in .
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 for , we can see condition (i) in Lemma 2.1 in [28] is met. Secondly, choose , and then 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 constant sufficiently small such thatand sufficiently big such thatNow we construct a continuous functionwhereSince is the minimum value point of function , then, . We construct a -function withwhere , , and will be determined as follows. here is used to make sure that is nonnegative. Therefore, the -function is nonnegative.
By using the Itô’s formula, we haveFrom system (23) and by using Itô’s formula, we haveIn view of the second equation on (23), we haveTherefore, we getSimilarly, according to the Itô’s formula, we haveNext, let us define the vector with elements . Since the generator matrix is irreducible, then for each , there exists one solution of the Poisson system (see Lemma 2.3 [29]) such thatwhere is the unit vector. Taking advantage of , one getsThen, we obtain thatIn the meanwhile, denote , and we haveBy using the Itô’s formula to , we obtainAccording (30) and (37)–(39), we havewhereAccording to condition , we derive thatIn the same way, we getLet us consider a compact subset as follows:where is small enough. Then we obtainIn summary, the conditions of Lemma 2.1 in [28] are fully satisfied, and then we conclude that diffusion process for system (23) is ergodic. Hence, stochastic process produces an ergodic stationary distribution. The proof is completed.
5. Extinction
Lemma 1 (see [30]). Let be a real-valued continuous local martingale vanishing at . Then,
The following lemma is essentially the same as that in [31], so we omit it.
Lemma 2. Let be the solution of system (3) with any initial value if the following hypothesis holds.
: there is a constant such thatthen the solution has the following properties:
Theorem 3. Let hypothesis hold. If , then the solution for stochastic system (4) with any initial value , will have the property thata.s., which implies the microorganism tends to extinction exponentially with probability one.
Proof. Respectively, from each equation of system (4), we can obtainAdd the left side of the three equations above to getwherethenIn the sequel,whereAccording to Lemma 2, we haveBy using the Itô’s formula and utilizing (54), we derive thatLet us divide by on both sides of the above formula, and we can getwhere satisfiesTogether with Lemma 1, we can getOn the other hand, from (10), we haveTake the limit on both sides of (58); if , then we haveHence, we getThis completes the proof.
6. Numerical Simulations
Now, we use numerical simulations to verify the results previously obtained. A discretization for system (3) implieswhere obeys the Gaussian distribution .
Firstly, we consider the cases of two regimes, without loss of generality, we let and , and the switching between them is governed by in the state space with the generator
A direct computation shows the stationary distribution is .
Example 1. For system (3), the parameters are chosen aswith the initial values .
A direct computation shows , 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).
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Example 2. For system (4), choose the parameters asand the initial values are same as Example 1.
By simple computation, we get , and according to Theorem 3, the microorganism will eventually become extinct (see Figure 3).
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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.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this study.
Acknowledgments
This work was supported by Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents.