Abstract and Applied Analysis

Volume 2014 (2014), Article ID 831573, 18 pages

http://dx.doi.org/10.1155/2014/831573

Research Article

Dynamics of a Stochastic Functional System for Wastewater Treatment

Xuehui Ji and Sanling Yuan

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

Received 20 January 2014; Accepted 2 February 2014; Published 24 March 2014

Academic Editor: Weiming Wang

Copyright © 2014 Xuehui Ji and Sanling Yuan. 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.

Abstract.

The dynamics of a delayed stochastic model simulating wastewater treatment process are studied. We assume that there are stochastic fluctuations in the concentrations of the nutrient and microbes around a steady state, and introduce two distributed delays to the model describing, respectively, the times involved in nutrient recycling and the bacterial reproduction response to nutrient uptake. By constructing Lyapunov functionals, sufficient conditions for the stochastic stability of its positive equilibrium are obtained. The combined effects of the stochastic fluctuations and delays are displayed.

1. Introduction

In the last few years, the use of mathematical models describing wastewater treatment is gaining attention as a promising method [1–6]. A basic chemostat model describing substrate-microbe interaction in an activated sludge process is as follows:

where and represent the concentrations of the substrate (biochemical oxygen demand) and microbes in an aeration tank at time , respectively. is the washout rate, is the input concentration of the substrate, and is the effective volume of the aeration tank; is the maximum uptake rate of the substrate; and are the half-saturation constants of the substrate and oxygen; respectively, is the decay rate of microbes and is the emission rate of the sludge; is the concentration of the dissolved oxygen and is a switching function describing the effect of on the uptake rate and the decay rate ; is the ratio of the concentration of mixed liquor suspended solids to the substrate. Some extensions and generalizations of the model have been proposed by many researchers (see [7–27], etc.).

Even though deterministic model (1) has a stable positive equilibrium under certain conditions, oscillations have been observed frequently in the growth of microbes during the experiments [28, 29], which have also been confirmed by many mathematical works for some extended chemostat models incorporating factors such as time delay [15–18, 30–32], periodic nutrient input [19–21, 33–35], feedback control [22–24], and stochastic environmental perturbations [25–27]. For a better understanding of microbial population dynamics in the activated sludge process, we take two steps towards developing model (1).

On the one hand, we take into account time delays that may exist in the process of wastewater treatment. By the death regeneration theory of Dold and Marais [36], the active biomass dies at a certain rate; of the biomass lost, the biodegradable portion adds to the slowly biodegradable organic matter which passes through the various stages to be utilised for active biomass synthesis, which requires some time for the completion of the regeneration. Also there is a time delay that accounts for the time lapse between the uptakes of substrates and the incorporation of these substrates, which has ever been observed from chemostat experiments with microalgae Chlamidomonas Reinhardii even when the limiting nutrient is at undetectable small concentration (see [37, 38], etc.). In the recent years, chemostat models with such time delays have been given much attention (see, e.g., [9, 14, 16–18, 39], etc.). In this paper, we will use distributed delays to describe the nutrient recycling and the time lapse between the uptakes of nutrient and the incorporation of this nutrient with delay kernels and , respectively.

On the other hand, in a real process of wastewater treatment there will be fluctuations in concentration of the substrate and microbe population due to stochastic perturbations from external sources such as temperature, light, and the like, or inherent sources in the chemical-physical and biological processes [40]. So we assume that model (1) is exposed by stochastic perturbations which are of white noise type and are proportional to the distances from values of the positive equilibrium , influence on the and , respectively. By this way, model (1) becomes in the following form:

where () are standard independent Wiener processes and () represent the intensities of the noises. is the fraction of the substrate regenerated from the dead biomass; is a general specific growth function.

Recently, stochastic biological systems and stochastic epidemic models have been studied by many authors; see, for example, Mao et al. [41, 42], Jiang et al. [43, 44], Liu and Wang [45, 46], and the references cited therein. But, as far as we know, there are few works on model (2). In this paper, our main purpose is to study the combined effect of the noises and delays on the dynamics of model (2), that is, whether and how the noises and delays affect the stability of . By the construction of appropriate Lyapunov functionals, we will show that the positive equilibrium keeps stochastically stable if the noises and delays are small. Furthermore, the sensitivities of the stability of with respect to the delays and noises are also discussed.

The paper is organized as follows. We first establish some preliminary results in Section 2. By constructing Lyapunov function(al)s, sufficient conditions for the stochastic stability of the positive equilibrium of the model without and with delays are obtained, respectively, in Sections 3 and 4. Numerical simulations and discussions are finally presented in Section 5.

2. Some Preliminaries

Define , , , , and . Then model (2) can be simplified as follows:

with initial value conditions

where , , the families of bounded continuous functions from to .

The corresponding deterministic model of (3) is

the special case of which when has ever been investigated by He et al. [18]. It is easy to see that model (5) has a positive equilibrium provided that

where

is globally asymptotically stable provided that the average delays are sufficiently small. Obviously, is still an equilibrium of stochastic model (3) if condition (6) holds.

We assume that function is nonnegative satisfying

And we extend the function by defining

so that is well defined in and is still of class in . Thus one can write

where represents terms of order in . Noting also that and , by condition (6), it follows that .

Introduce new variables , ; then model (3) can be rewritten as follows:

where

Note that if , then model (11) has the form

where

Obviously, model (13) has the same equilibrium as model (11), and the stochastic stability of the positive equilibrium of model (3) is equivalent to the zero solution of model (11). We wonder how the stochastic perturbations and delays affect the dynamics of model (3) or (11).

Before starting our analysis, we first give some basic theories in stochastic differential equations and stochastic functional differential equations [47–49]. 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 Brownian motions defined on this probability space. Consider the following -dimensional stochastic differential equation:

Definition 1. The trivial solution of system (15) is said to be as follows:

(i)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)stochastically asymptotically stable if it is stochastically stable and, moreover, for every , there exists a such that

 whenever ,(iii)globally asymptotically stable in probability if it is stochastically asymptotically stable and, moreover, for all

Lemma 2. If there exists a nonnegative function , two continuous functions , , and a positive constant such that, for ,

hold.

(i)If

 then the trivial solution of system (A.1) is stochastically stable.(ii)If there exists a continuous function such that

 holds, then the trivial solution of system (15) is stochastically asymptotically stable.(iii)If (ii) holds and moreover

 then the trivial solution of system (15) is globally asymptotically stable in probability.

For the stability of the equilibrium of a nonlinear stochastic system, it can be reduced to problems concerning stability of solutions of the linear associated system. The linear form of (15) is defined as follows:

Lemma 3. If the trivial solution is stochastically stable for the linear system (23) with constant coefficients (, ) and the coefficients of systems (15) and (23) satisfy the following inequality:

in a sufficiently small neighborhood of , with a sufficiently small constant , then the trivial solution of system (15) is asymptotically stable in probability.

Consider the following -dimensional stochastic functional differential equation

with initial condition , where is the space of -adapted random variables , with for , and

Definition 4. The trivial solution of system (25) is said to be

(i)mean square stable if, for each , there exists such that for any initial process ,

 for any provided that ,(ii)asymptotically mean square stable if it is mean square stable and

(iii)stochastically stable if for any and , there exists a such that

 provided that .

3. Dynamical Behavior of the System without Delays

We first study the stochastic stability of the equilibria of model (13). Throughout the paper, we assume that the basic hypotheses given in the Section 2 are satisfied. The linearized system of model (13) is

For convenience, let

For linearized system (30), we have the following theorem.

Theorem 5. Let condition (6) hold. If

then the trivial solution of system (30) is globally asymptotically stable in probability.

Proof. Define a smooth function by

Then using Itô’s formula, for all , we have

where

By (31), we obtain

We take () by

thus the thesis follows by Lemma 2. This completes the proof of Theorem 5.

Now, we are in a position to prove the stability of the trivial solution of model (13).

Theorem 6. Let condition (6) hold. If the conditions in (32) are satisfied, then the trivial solution of model (13) is stochastically asymptotically stable.

Proof. For a sufficiently small constant , , we have

Note that are the terms of order in and ; then we have

Thus for a sufficiently small constant , we have

provided . Therefore,

Applying Lemma 3 and Theorem 5, we obtain the conclusion.

4. Dynamical Behavior of the System with Delays

We now study the stability in probability of the equilibria of system (11). Its corresponding linearized system is

Define the average time lags as

and let , be defined in (31). For linearized system (42) we have the following theorem.

Theorem 7. Let condition (6) hold. If

then the trivial solution of system (42) is asymptotically mean square stable.

Proof. Consider the function defined in (33). It follows from (42) and Itô’s formula that

Straightforward computations lead to

From the terms of the right-hand side of (46), we have

For the term , it is clear that

where

For the term , we have that

where

Substituting (47)–(48) together with (51) into (46), we obtain

For technical reasons, we assume that and . Then the function

is well defined. Using Itô’s formula, we have

We now consider the function

It follows from (56) and (57) that

Therefore, for the function

we have

By (44), we choose such that

Let such that and for all . Then for all , one has

For convenience, let

Integrating both sides of (62) from to , we have

Discussing as that in He et al. [18], by the Barbălat lemma, we conclude as . Applying Definition 4, we obtain the conclusion.

Now, we are in a position to prove the stability of the trivial solution of nonlinear system (11) using the Lyapunov functionals constructed above.

Theorem 8. Let condition (6) hold. If conditions (44) are satisfied, then the trivial solution of the system (11) or the equilibrium of system (6) is stochastically stable.

Proof. Consider the Lyapunov function defined in (33). It follows from (11) and Itô’s formula that

where

From the terms of the right-hand side of (66), we observe that

where is defined in (49), and

where is defined in (52). Substituting (67) and (68) into (46), we get

For the functions and defined in (55) and (57), one has

It follows from the expression of and that

For , one has

Since and are terms of order in , , then we have

For , we can find a constant such that

provided that . Now consider the class of processes

Notice that for ,

are valid. Substituting (74)-(76) into (72), we obtain

Integrating both sides of the above formula from to yields

where

By the definition of function , we can find a constant such that

Obviously,

where . Now for , let

and . Then it follows that

On the other hand, we have

Hence, we have . Let ; then

Equivalently,

Applying Definition 4, we obtain the conclusion.

5. Simulations and Discussions

In this paper, we have considered a stochastic chemostat model simulating the process of wastewater treatment. The model incorporates a general nutrient uptake function and two distributed delays. The first delay models the fact that nutrient is partially recycled after the death of the biomass by bacterial decomposition and the second indicates that the growth of the species depends on the past concentration of the nutrient. Furthermore, we consider the stochastic perturbations which are of white noise type and are proportional to the distances of , from the values of the positive equilibrium , . By constructing appropriate Liapunov-like functionals, some sufficient conditions for the stochastic stability of the positive equilibrium have been obtained.

For model (3), we have first analyzed the stochastic stability of the positive equilibrium in the case when the delays are ignored, that is, the average delays . Our findings in Theorem 6 reveal that is stochastically stable provided that the intensities of noises are small. When at least one of the average delays and is not equal to zero, our results in Theorem 8 reveal that is stochastically stable provided that the average delays and are both small. Obviously, Theorem 8 reduces to Theorem 6 when , which indicates that if the average delays are sufficiently small, is still stochastically stable; and in the case of (), Theorem 8 reduces to He et al. [18, Theorem 3.1]; that is to say, the equilibrium of model (3) is still stable if and are sufficient small, which preserves the dynamics of its corresponding deterministic counterpart (5).

To illustrate the results obtained above, some numerical simulations are carried out by using Milstein scheme [50]. Here we assume that the specific growth function is of Michaelis-Menten type

where is the half-saturation constant. For the kernel functions and , we consider two special cases: (1) ; (2) and . For case (1), the discretization of model (3) for takes the form

where time increment and is -distributed independent random variables which can be generated numerically by pseudorandom number generators. For case (2), define

then the discretization of model (3) for , takes the form

Let in model (3) , , , , , , . It is easy to compute that , , , , and .

The first two examples given below concern case (1) when the delays are ignored; that is to say, it is assumed that the process of nutrient recycling and the growth response of the species are immediate and, therefore, . Example 1 verifies the results obtained in Theorem 6.

Example 1. Let and , then by straightforward computations, we have that , . In view of Theorem 6, the equilibrium of (3) is stochastically asymptotically stable, which is consistent with the simulation results as shown in Figure 1.

Figure 1: The dynamics of stochastic model compared with deterministic model with and . Here , .

To further study the combined effects of , when , we need to consider four situations: (a) increases, increases; (b) increases, decreases; (c) decreases, increases; (d) decreases, decreases. Here we only give one example about situation (a); other situations can be considered similarly.

Example 2. Let the intensities , increase from , to , , respectively. Simulations show that the trajectories of model (3) still approach ultimately to the positive equilibrium , but they need to go through more oscillations and more time to return to (see Figure 2).

Figure 2: The dynamics of stochastic model with different values of and . Here , .

The next two examples concern case (2) when and take weak kernels; that is, and , which means that and . Example 3 verifies the results obtained in Theorem 8.

Example 3. Let , , and . It is easy to compute that , + and , ; thus conditions (44) are satisfied. By Theorem 8, the equilibrium of model (3) is stochastically stable. Our simulation supports this result as shown in Figure 3.

Figure 3: The dynamics of stochastic functional model with different . Here , , , .

To examine the combined effects of the noise intensities and the delays on the dynamics of model (3), we first consider the case when the values of in Example 3 are fixed and the values of and are reduced from and to and , respectively. That is to say, the average delays and increase from and to and , respectively. Simulation results show that the solution of (3) will suffer more oscillations and more time to approach the equilibrium when delays increase (see Figure 3). When both the values of the noise intensities and the delays vary, the dynamics of model (3) may become more complicated. Here we only consider the case when (), and (i.e., and ) all increase. See the following Example.

Example 4. Let (), and (i.e., and ) increase from 0.1, 0.08, 1, and 0.2 (i.e., and ) to 1, 0.8, 10, and 10 (i.e., and ), respectively. It is found that the trajectories of model (3) fluctuate wildly and suffer more oscillations and need more time to approach the equilibrium ; please see Figure 4.

Figure 4: The dynamics of stochastic functional model with different and . Here , , , .

Notice also that conditions (44) in Theorem 8 are only sufficient conditions to insure the stochastic stability of , which are dependent on parameters , , , and . Define

Thus, conditions (44) are equivalent to those when parameters , , , and are seated in the following parameter set:

from which we can further perform some approximate sensitivity analysis of the stochastic stability of with respect to these parameters. To do this, we can let two of the parameters (e.g., and ) vary and the other two ( and ) be fixed, which have six cases in all.

Let us first consider the case when and ; then and are both functions of and . Then defined in (92) is equivalent to

which is the projection of surfaces and in the first octant such that (see Figure 5). The positive equilibrium is stochastically stable provided that .

Figure 5: The positive equilibrium is stochastically stable provided that . Here and .

To better observe the dependence of the stochastic stability of on all parameters, we further consider another two cases when and are fixed and and are fixed. Accordingly, defined in (92) is equivalent, respectively, to

which are plotted, respectively, in Figures 6 and 7 (other three cases can be considered similarly). From Figures 5–7, we find that the stochastic stability of is greatly affected by , and and less affected by (which is consistent with the results observed in [13, 17]). We would like to point out here that may also be stable when the parameters are seated outside of the set , since (44) are only sufficient conditions ensuring the stochastic stability of .

Figure 6: The positive equilibrium is stochastically stable provided that . Here and .

Figure 7: The positive equilibrium is stochastically stable provided that . Here and .

In conclusion, this paper presents an investigation on the combined effect of the noises and delays on a bottom-microbe model. Our findings are useful for better understanding of the dynamics of microbial population in the activated sludge process. We should point out that there are still some other interesting topics about the wastewater treatment deserving further investigation, for example, membrane reactor, and so forth. We leave these for future considerations.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 11271260), Shanghai Leading Academic Discipline Project (no. XTKX2012), and the Innovation Program of Shanghai Municipal Education Commission (no. 13ZZ116).

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