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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 423154, 11 pages
Asymptotic Behavior of a Chemostat Model with Stochastic Perturbation on the Dilution Rate
1College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
2Mathematics, FEIS, Swinburne University of Technology, P.O. Box 218 (H38) Hawthorn, VIC 3122, Australia
Received 4 October 2012; Revised 1 January 2013; Accepted 4 January 2013
Academic Editor: Ivanka Stamova
Copyright © 2013 Chaoqun Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We present a stochastic simple chemostat model in which the dilution rate was influenced by white noise. The long time behavior of the system is studied. Mainly, we show how the solution spirals around the washout equilibrium and the positive equilibrium of deterministic system under different conditions. Furthermore, the sufficient conditions for persistence in the mean of the stochastic system and washout of the microorganism are obtained. Numerical simulations are carried out to support our results.
Modeling microbial growth is a problem of special interest in mathematical biology and theoretical ecology. One particular class of models includes deterministic models of microbial growth in the chemostat. Equations of the basic chemostat model with a single species and a single substrate take the form where and , respectively, denote concentrations of the nutrient and the microbial biomass, and all the parameters are positive constants; denotes the feed concentration of the nutrient and denotes the volumetric dilution rate (flow rate/volume). The function denotes the microbial growth rate and a typical choice for is . The stoichiometric yield coefficient denotes the ratio of microbial biomass produced to the mass of the nutrient consumed.
The dynamics of the basic model is simple. If is constant and is a monotonically increasing function, then the microorganism can either become extinct or persist at an equilibrium level [2–5]. The particular outcome depends solely on the break-even concentration where . Specifically, if , the organism persists, and if , it becomes extinct.
Note that the modeling process that leads to (1) relies on the fact that the stochastic effects can be neglected or averaged out, thanks to the law of large numbers. This is possible only at macroscopic scale, for large population sizes, and under homogeneity conditions. At all other scales or when the homogeneity conditions are not met, random effects cannot be neglected, just as that stated in Campillo et al. : “This is the case at microscopic scale, in small population size, as well as all scales preceding the one where (1) is valid. This is also when the homogeneity condition is not met, for example, in unstirred conditions. Also the accumulation of small perturbations in the context of multispecies could not be neglected. Moreover, whereas the experimental results observed in well mastered laboratory conditions match closely the ODE theoretical behavior, a noticeable difference may occur in operational.” So, even if the description (1) is sufficient for a number of applications of interest, it does not account for the stochastic aspects of the problem (see Campillo et al.  for more details on this respect).
In fact, ecosystem dynamics is inevitably affected by environmental white noise which is an important component in realism. It is therefore useful to reveal how the noise affects the ecosystem dynamics. As a matter of fact, stochastic biological systems and stochastic epidemic models have recently been studied by many authors; see [7–16]. We also refer the readers to Imhof and Walcher  for the reason why the stochastic effects should be considered in chemostat modeling.
Taking into account the effect of randomly fluctuating environment, we introduce randomness into model (1) by replacing the dilution rate by , where is a white noise (i.e., is a Brownian motion) and represents the intensity of noise. This is only a first step in introducing stochasticity into the model. Ideally we would also like to introduce stochastic environmental variation into the other parameters such as the feed nutrient concentration and the microbial growth rate , but to do this would make the analysis too difficult. As a result, model (1) becomes the following system of stochastic differential equation:
To the best of our knowledge, a very little amount of work has been done with the above model. Motivated by this, in this paper, we will investigate the long time behavior of model (2) with taking the Holling II functional response function, that is, .
We should point out that a few papers have already addressed the stochastic modeling of the chemostat [6, 7, 17–19]. Here we only mention a recent paper by Imhof and Walcher . They introduced a variant of the deterministic single-substrate chemostat model for which the persistence of all species is possible. To derive a stochastic model they considered a discrete-time Markov process with jumps corresponding to the deterministic system added with a centered Gaussian term, letting the time step converges to zero leads to a system of stochastic differential equations. They proved that random effects may lead to extinction in scenarios where the deterministic model predicts persistence; they also established some stochastic persistence results. Obviously, the model they considered is different from our model (2). We also refer the readers to Campillo et al.  for other works on the stochastic modeling of the chemostat.
The organization of this paper is as follows. The model and some basic results about the model are presented in the next section. In Section 3, the long time behavior of the stochastic model, including the asymptotic behavior around the positive equilibrium point and that around the extinction equilibrium point, is analyzed. An extinction result on the model is presented in Section 4. Finally, in Section 5, simulations and discussions are presented.
2. The Model and Some Fundamental Results
Assuming in model (2) that takes the Holling II functional response function, that is, , consider the following stochastic chemostat model: where and have the similar biological meanings as in model (1), and all parameters are positive constants. , , and play similar roles as in model (2), and and are the maximal growth rates of the organism and the Michaelis-Menten (or half saturation) constant, respectively; the yield coefficient has been scaled out by scaling.
As and in (3) are concentrations of the substrate and the microorganism at time , respectively, we are only interested in the positive solutions. Moreover, in order for a stochastic differential equation to have a unique global (i.e., no explosion in a finite time) solution for any given initial value, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition. However, the coefficients of (3) do not satisfy the linear growth condition, though they are locally Lipschitz continuous, so the solution of model (3) may explode at a finite time [20, 21]. In the following, by using the comparison theorem for stochastic equation (see [8, 22]), we first show the solution of model (3) is positive and global.
Theorem 1. For any given initial value , there is a unique positive solution to model (3) on , and the solution will remain in with probability one, namely, for all almost surely.
Proof. Since the coefficients of model (3) are locally Lipschitz continuous for any given initial value , there is a unique positive local solution on , where is the explosion time (i.e., the moment at which the solution tends to infinity). Next, we show that .
Since the solution is positive, we have
Denote by the solution of the following stochastic differential equation: and the solution of the equation
By the comparison theorem for stochastic equations, we have , , and , a.s.
Similarly, we can get
Denote by the solution of stochastic differential equation
and the solution of the equation
We have that , , , a.s.
To sum up, we have that
Noting that (5)–(9) are all linear stochastic differential equations, , , , and can be explicitly solved from them, separately. Obviously, they are all positive and globally existent for all . From (10), we can have that . The proof is thus completed.
The following theorem shows that the solution of model (3) with any positive initial value is uniformly bounded in mean.
Theorem 2. For any given initial value , the solution of model (3) has the property:
Proof. Define the function , by the Itô formula, we get
Integrating both sides from 0 to , and then taking expectations, yields
It is clear that
Thus, we complete the proof of Theorem 2.
3. The Long Time Behavior of Model (3)
When , model (3) becomes its corresponding deterministic system
We have known that the dynamic behavior of model (16) is completely determined by the break-even concentration . Model (16) always has a washout equilibrium . When , is globally asymptotically stable, and when , loses its stability and a globally asymptotically stable positive equilibrium appears, where and .
When , is still an equilibrium of model (3) but is not; that is to say, model (3) has no positive equilibrium. It is natural to ask whether the microorganism will persist or go to extinction in the chemostat. In this section we mainly use the way of estimating the oscillation around (or ) to reflect how the solution of model (3) spirals around (or ). In the following part of this section, we always denote by
3.1. Asymptotic Behavior around the Washout Equilibrium
As mentioned above, system (16) always has an equilibrium which is globally asymptotically stable provided . It is natural to ask whether the solutions of system (3) will be close to . We have the following theorem.
Theorem 3. If and , then the washout equilibrium of system (3) is stochastically asymptotically stable in the large.
Proof. Define a function by
Obviously, the function is positive definite, and where,
We will show that for all solutions of system (3) with initial value ,
Suppose first that , then
It then follows from (20) that
Suppose next that ; noting also that , we have
It follows from (20) that
To sum up, (21) holds. That is to say, is negative definite. By Theorem A in the appendix, the washout equilibrium of system (3) is stochastically asymptotically stable in the large. The proof of Theorem 3 is thus completed.
3.2. Asymptotic Behavior around the Positive Equilibrium
Theorem 4. If and , then for any solution of system (3) with initial value ,
Proof. The proof is inspired by the method of Imhof and Walcher . Define a function by
Obviously, the function is positive definite. Using Itô's formula we get where
We will show that for all solutions with initial value ,
Suppose first that , then and
By (31) we have
Suppose next that and (which implies that , ), then
By (31), we have
Finally, if and , then
By (31) we have
To sum up, (32) holds. Therefore,
Integrating both sides from to yields where . Obviously, is a local continuous martingale. From (3), we can get It follows that where . Thus we have that which implies that
By Strong Law of Large Numbers (see Mao ), we obtain
It follows from (41) that namely,
This completes the proof of Theorem 4.
3.3. Persistence of System (3)
From the result of Theorem 4, we conclude that system (3) is persistent when is small, which implies the persistence of microorganism. L. S. Chen and J. Chen in  proposed the definition of persistence in the mean for the deterministic system. Here, we also use this definition for the stochastic system (see also Ji et al. ).
Definition 6. System (3) is said to be persistent in the mean, if
4. Washout of the Organism in the Chemostat
The following theorem shows that sufficiently large noise can make the microorganism extinct exponentially with probability one in a simple chemostat.
Theorem 8. For any given initial value , the solution of system (3) has the property:
Proof. Define the function ; by the Itô formula, we get
Integrating both sides from 0 to yields
Dividing on the both sides and letting , we have
The proof of Theorem 8 is completed.
Remark 9. In fact, we can see from the proof of Theorem 8 that only the boundedness of is used. So, Theorem 8 holds true for a much larger class bounded growth rate functions, for example, (see also Figure 4).
5. Simulations and Discussions
In order to confirm the results above, we numerically simulate the solution of system (3) and system (16) with the initial . The numerical simulation is given by the following Milstein scheme . Consider the discretization of system (3) for : where time increment , and is -distributed independent random variables which can be generated numerically by pseudorandom number generators. In Figures 1–4, we will use the blue lines and the red lines to represent the solutions of deterministic system (16) and those of stochastic system (3), respectively.
By Theorem 3, we expect that the washout equilibrium of system (3) is globally asymptotically stable under the conditions and . In Figure 1, we choose parameters ,, , , and in Figure 1(a). We can compute that and . Then the washout equilibrium of systems (3) and (16) is globally asymptotically stable (see Figure 1(a)). Furthermore, to watch the influence of the intensity of noise on the dynamics of system (3), we take in Figure 1(b). We can see that the solutions of system (3) with tend to the washout equilibrium faster than that of system (3) with .
Next, we consider the effect of stochastic fluctuations of environment on the positive equilibrium of the corresponding deterministic system. As mentioned in the Section 2, there is a positive equilibrium of system (16) when , and it is globally asymptotically stable. Besides, Theorem 4 tells us that the difference between the perturbed solution and is related to white noise under the conditions and . In systems (3) and (16), we choose parameters , , , and . As for the washout equilibrium , we take in Figure 2(a) and in Figure 2(b). We can compute that and in Figure 2(a) and in Figure 2(b). As expected, the solutions of system (3) are oscillating around the positive equilibrium for a long time (see Figure 2). Besides, we can observe that with white noise getting weaker, the fluctuation around gets smaller, which supports the result of Theorem 4.
Theorem 8 shows that the microorganism will be washed out eventually under the condition even if system (16) has a positive equilibrium. As an example, we choose parameters , , , and in systems (3) and (16). We can compute that ; thus system (16) will persist. But when is large enough, for example, we take , we can compute that , and the microorganism will die out (see Figure 3). This shows that strong noise may lead to the extinction of the species in the chemostat.
To further confirm that Theorem 8 holds for a much lager class of growth rate function (i.e., Remark 9), we take as the Holling IV functional response function instead of and all parameters have the similar values as above except for in systems (3) and (16). Numerical simulations show that they have the similar dynamics as for (see Figure 4).
Some interesting questions deserve further investigation. One may propose more realistic but complex models, such as incorporating the colored noise into the system. Moreover, it is interesting to study other parameters perturbation.
For the completeness of the paper, we list some basic theory in stochastic differential equations (see [21, 26, 27]). Let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null sets). Let be the standard Brownian motions defined on this probability space. Denote
In general, consider -dimensional stochastic differential equation with initial value . Assume that the assumptions of the existence-and-uniqueness theorem are fulfilled. Hence, for any given initial , has a unique global solution that is denoted by . Assume furthermore that
If acts on a function , then
Definition A. (i) The trivial solution of is said to be stochastically stable or stable in probability if for every pair of and , there exists a such that
whenever . Otherwise, it is said to be stochastically unstable.
(ii) The trivial solution is said to be stochastically asymptotically stable if it is stochastically stable and, moreover, for every , there exists a such that whenever .
(iii) The trivial solution is said to be stochastically asymptotically stable in the large if it is stochastically asymptotically stable and, moreover, for all ,
Theorem A. If there exists a positive-definite decreasing radially unbounded function , such that is negativedefinite, then the trivial solution of is stochastically asymptotically stable in the large.
The authors are grateful to the handling editor and anonymous referees for their careful reading and constructive suggestions which lead to truly significant improvement of the paper. Research is supported by the National Natural Science Foundation of China (11271260, 11001212, and 1147015) and the Innovation Program of Shanghai Municipal Education Commission (13ZZ116).
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