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.
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.where and represent the concentration of nutrient and microorganisms at the time , separately. expresses the input concentration of nutrition, is Brownian motion defined on a complete probability space () and stands for the density of the white noise, ( and are constants) denotes the dilution rate of the turbidostat and ( is constant) is the sum of the dilution rate and death rate of the microorganisms in the system. is the Holling 2 functional response function ( and are constants).
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.
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. If , then stochastic system (1) has a unique positive solution almost surely with initial value .
Proof. For , letAccording to the It formula, we havewith initial value . It is obvious that and are continuous and differentiable functions. Hence (3) satisfies local Lipschitz condition, which means that there exists a unique local solution for . Then is the unique positive local solution for system (1) with the initial value . Next, we prove that the positive solution of system (1) is global; that is, . We first define the filtration as and choose an enough large such that . Then for all , we define a stopping time byNext, we further show . According to the definition of the stopping time, Therefore, is increasing as . Set . Then . Now, it is sufficient for us to show . Arguing by contradiction, we assume that . Then there exist constants and such that . Therefore, there is a constant such that for all .
Define a function bywhere is a positive constant. It is obvious that . Hence . By the It formula, the following equation can be derivedwhere Through calculations, the following inequality can be obtained Defining , we have which yields Integrating from to and taking expectation in both sides of inequality (11), one can obtainLet for all , then . Based on the definition of the stopping time, we have or for all . Hence there is a positive constant such that . Furthermore, Setting , we have , which is contradictory with . Hence . This completes the proof.
Remark 2. If we take as a threshold value, then the microorganisms in system (1) will survive as and be extinct as .
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
Lemma 3. The solution of system (1) satisfies with any initial value . Here, is a positive constant satisfying
Proof. Let and where is a nonnegative constant decided later. By the It formula, we derive We can further obtain Denote where Let be a constant. Then For , we have where Define a function by and which gives Integrating from to and taking expectation for (25), one has Then it is easy to obtain which leads to is a continuous function, so there is a constant such that According to (24), for small enough , , integrating from to and taking expectation, we can get where here, Applying Burkholder-Davis-Gundy inequality [41], we have Hence, we have Particularly, one can choose such that which yields For any , the following inequality can be obtained by Chebyshev’s inequality [41], Using Borel-Cantelli Lemma [41], for all and sufficiently large , we have Therefore, there is a , for all and ; the above equation holds. In addition, for all , when and , which means When , where is determined by (22). Therefore, This completes the proof.
Lemma 4. If , then the solution of system (1) with initial value satisfies where is the break-even concentration of and is the equilibrium concentration of .
Proof. Define It follows from , Lemma 3, and Burkholder-Davis-Gundy inequality [41] that By Doob’s martingale inequality [41], for arbitrary , it follows that By Borel-Cantelli Lemma [41], for all and sufficient large , we derive Thus, there exists a positive constant such that for all and , which implies Taking superior limit leads to Sending , we have Then there exist a constant and a set for small enough such that . Thus for , we have which means Together with we derive and Applying the same methods, we can prove that This completes the proof.
Definition 5. Stochastic equation has a solution weakly converging to the distribution . Here is a probability measure of such that . Particularly, has density , where is a normal constant and
Theorem 6. is the solution of system (1) with any initial value . Moreover, if then we have where , , is the break-even concentration of , is determined later by (74), is the boundary equilibrium of the corresponding deterministic model, and is defined in Definition 5.
Proof. The deterministic system corresponding to (1) has an equilibrium under the condition of , where and satisfy Define function by Then by the It formula, we get where According to (66), we have It follows from some calculations that Set then where Integrating (68) from to and dividing by on both sides, one can obtain According to Lemma 4, it follows thatDefine Using the It formula, we have Then integrating (78) from to on both sides and dividing all by , we can get which means Based on (76), and , we know that If then we have that is, 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 It formula, it follows that Integrating the above two equations from to on both sides and dividing them by , we obtain Therefore, which implies thatNext, applying the It formula to , one deduces Integrating the above inequality from to on both sides and dividing all by , we have According to Lemma 3, we have which yields that is, Therefore, if then This completes the proof.
Theorem 7. System (1) with admits the positive solution . Furthermore, if then and weakly converges to distribution ; here is defined as in Definition 5, which means the biomass of all species in the vessel will be extinct exponentially.
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. Applying the It formula to , we derive that Since is a monotonous increasing function and , Using the conditions that , and we know that which means tends to zero exponentially almost surely.
Now we prove that (62) is stable in distribution. Set , then we have According to Theorem 2.1 in [42], we know that diffusion process is stable in distribution when . Therefore, is also stable in distribution. We further prove that is stable in distribution as . We define the following stochastic process with the initial value by Then The solution of (103) is According to Theorem 6, when , , and . Therefore, for almost all , such that In addition, for all , if , it follows that and when , which implies thatNow, we consider Integrating it from to , one can derive Taking expectation for both sides of the above inequality, we obtain then which leads to According to (108), (113), and , we can get Therefore, based on the fact that is stable to distribution as , weakly converges to distribution . 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 in the vessel will be extinct and will survive due to the fact that is inadequate competitor [2] if .
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 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.
Let , , , , , , , , , , , , and , then system (1) becomesWe derive , , , and . Obviously, . The concentration of nutrient at equilibrium is . Hence, the species besides 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 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 and microorganisms in the turbidostat. The stochastic factors always cause the fluctuation of and (see Figures 1–5).
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Although the concentration of nutrients varies significantly in the system, it always fluctuates around (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 fluctuates around (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 and . The species will be extinct since the concentration of nutrient is less than its break-even concentration (see Figure 1).
To further confirm the results obtained in Theorem 7, we give the following example.
Set , , , , , , , , , , , , and , then system (1) arrives atIt follows from simple calculations to obtain , which satisfies the conditions of Theorem 7. Hence, and tend to zero exponentially, which implies , (see Figure 6).
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In fact, if the density of white noise increased (such as extremely high temperature) so that ; 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.
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 paper.
Acknowledgments
We are very grateful to Prof. Sigurdur F. Hafstein of University of Iceland who offered invaluable assistance. This work was supported by the National Natural Science Foundation of China (grant numbers 11561022 and 11701163) and the China Postdoctoral Science Foundation (grant number 2014M562008).