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 [

Even though deterministic model (

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 [

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 [

Recently, stochastic biological systems and stochastic epidemic models have been studied by many authors; see, for example, Mao et al. [

The paper is organized as follows. We first establish some preliminary results in Section

Define

The corresponding deterministic model of (

We assume that function

Introduce new variables

Before starting our analysis, we first give some basic theories in stochastic differential equations and stochastic functional differential equations [

The trivial solution of system (

stochastically stable or stable in probability if for every pair of

whenever

stochastically asymptotically stable if it is stochastically stable and, moreover, for every

whenever

globally asymptotically stable in probability if it is stochastically asymptotically stable and, moreover, for all

If there exists a nonnegative function

If

then the trivial solution of system (A.1) is stochastically stable.

If there exists a continuous function

holds, then the trivial solution of system (

If (ii) holds and moreover

then the trivial solution of system (

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 (

If the trivial solution is stochastically stable for the linear system (

Consider the following

The trivial solution of system (

mean square stable if, for each

for any

asymptotically mean square stable if it is mean square stable and

stochastically stable if for any

provided that

We first study the stochastic stability of the equilibria

Let condition (

Define a smooth function

Now, we are in a position to prove the stability of the trivial solution

Let condition (

For a sufficiently small constant

We now study the stability in probability of the equilibria

Let condition (

Consider the function

Now, we are in a position to prove the stability of the trivial solution

Let condition (

Consider the Lyapunov function

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

For model (

To illustrate the results obtained above, some numerical simulations are carried out by using Milstein scheme [

Let in model (

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,

Let

The dynamics of stochastic model compared with deterministic model with

To further study the combined effects of

Let the intensities

The dynamics of stochastic model with different values of

The next two examples concern case (2) when

Let

The dynamics of stochastic functional model with different

To examine the combined effects of the noise intensities and the delays on the dynamics of model (

Let

The dynamics of stochastic functional model with different

Notice also that conditions (

Let us first consider the case when

The positive equilibrium

To better observe the dependence of the stochastic stability of

The positive equilibrium

The positive equilibrium

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.

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

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).