Advanced Nonlinear Dynamics of Population Biology and EpidemiologyView this Special Issue
Dynamics of a Stochastic Functional System for Wastewater Treatment
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.
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 , 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 . 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. . 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.
Now, we are in a position to prove the stability of the trivial solution of model (13).
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.
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. , 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.
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