Dynamics of a stochastic eutrophication-chemostat model with impulsive dredging and pulse inputting on environmental toxicant

In this paper, we present a stochastic eutrophication-chemostat model with impulsive dredging and pulse inputting on environmental toxicant. The sufficient condition for the extinction of microorganisms is obtained. The sufficient condition for the investigated system with unique ergodic stationary distribution is also obtained. The results show that the stochastic noise, impulsive dredging, and pulse input on the environmental toxicant play important roles in the extinction of microorganisms. The results also indicate the effective and reliable controlling strategy for water resource management. Finally, numerical simulations are employed to illustrate our results.


Introduction
The chemostat is a device for continuous and impulsive cultures of microorganisms in laboratory [1][2][3]. Impulsive differential equations are found in almost every domain of applied science and have been studied in many investigations [4,5]. With the development of society, the increasing amount of toxicants and contaminants have entered ecological systems. Environmental pollution has become one of the most important society-ecological problems. Therefore, it is very important to study the effects of toxicants on a population or community. Specially, the toxicant and abundant microorganisms in the water pollution environment are also a threat to the water resource management. Consequently, it is important to discuss chemostat models in a polluted environment [6,7]. Zhou et al. [8] considered that reservoir dredging is the main and effective way to improve water quality by using a physical method. However, it is well known that many real-world systems may be disturbed by stochastic factors. Population systems are often subjected to various types of environmental noise. In ecology, it is critical to discover whether the presence of this noise has significant effects on population systems. Mao [9,10] investigated stochastic differential equations and their applications. Lv et al. [11] presented an impulsive stochastic chemostat model with nonlinear perturbation.

A+x(t)+By(t)
-Dy(t)rc o (t)y(t)] dt + y(t)(σ 21 + σ 22 y(t)) dB 2 (t), dc o (t) = (fc e (t) -(g + m)c o (t)) dt, dc e (t) = (-hc e (t)) dt, ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ t = (n + l)τ , t = (n + 1)τ , where x(t) is the concentration of the nutrient in a lake at time t. y(t) is the concentration of the microorganism in a lake at time t. c o (t) is the concentration of the toxicant in the organism of the microorganism in a lake at time t. c e (t) is the concentration of the toxicant in a lake at time t. D denotes the input rate from the lakes containing the nutrient and the wash-out rate of nutrients and microorganisms from the lake. β > 0 is the uptake constant of the nutrient.
x(t) A+x(t)+By(t) is a functional response of the Beddington-DeAngelis type. k > 0 is the yield of the microorganism y per unit mass of the nutrient. A > 0 and B > 0 are the saturating parameters of the Beddington-DeAngelis functional response. r > 0 is the depletion rate coefficient of the microorganism y due to the microorganism organismal toxicant. f > 0 is the coefficient of the population organism's net uptake of toxicant from the environment in a lake. -g < 0 and -m < 0, respectively, represent coefficients of the elimination and depuration rates of the toxicant in the organism in a lake. -h < 0 is the coefficient of the totality of toxicant losses from the system environment in a lake, including processes such as biological transformation, chemical hydrolysis, volatilization, microbial degradation, and photosynthetic degradation. τ is the period of impulsive dredging or the pulse input environmental toxin. 0 < h 1 < 1 is the effect of impulsive dredging microorganism at time t = (n+l) (0 < l < 1). 0 < h 2 < 1 is the effect of impulsive dredging environmental toxicant at time t = (n + l) (0 < l < 1). μ ≥ 0 is the amount of pulse input of environmental toxin concentration in a lake at t = (n + 1)τ , n ∈ Z + , and Z + = {1, 2, . . .}.
We can easily have a unique fixed point (c * o , c * e ) of system (3.3) as follows:  , c e (t)), which is also globally asymptotically stable, c o (t) and c e (t) are defined as follows: , where c * o , c * e are defined as (3.4), and , (3.6) where c * o and c * e are defined as (3.4), and c * * o and c * * e are defined as (3.6). For convenience, we consider the following notation: Define (x(t), y(t)) and (w(t), z(t)) are the solutions of the subsystem of system (2.1), respectively: 8) and the following SDE without impulsive perturbations: with the initial value w(0) = x(0) and z(0) = y(0).

Lemma 3.3
The solutions (x(t), y(t)) of the subsystem of system (2.1) can also be expressed as follows: where (w(t), z(t)) is the solution of (3.10).
Proof One can find that (x(t), y(t)) is continuous on the interval (τ n , τ n+l ), and for t = τ n+l , (3.11) For every n ∈ N , and τ n+l ∈ [0, +∞), and Assumption 3.4 is a general assumption which is the condition for Lemma 3.6.

Lemma 3.5 ([12]) If Assumption 3.4 holds, the Markov process X(t) has a stationary distribution μ(·), and
where f is an integrable function with respect to the measure μ.

The dynamics
In the following theorem, we devote ourselves to investigating system (3.10). where for x ∈ (0, +∞) and constant C satisfies that Proof Constructing the following auxiliary differential equation: with the initial value W (0) = x(0) > 0, we assume that W (t) is the solution of (4.3). Obviously, the following inequality can be obtained by the comparison theorem for stochastic differential equations: We set and compute the following indefinite integral: Applying Itô's formula, we have Integrating with respect to t from 0 to t on both sides of (4.10), we have where M(t) = σ 22 0<τ n+l <t (1h n+l ) t 0 z(s) dB 2 (t) and its quadratic variation is given by According to the exponential martingales inequality, for any positive τ , α, β, (4.14) There exists random k 0 1 ∈ k 1 (ω) such that k 1 > k 0 1 for almost all ω ∈ Ω. We can obtain the following by the Borel-Cantelli lemma: Then, for 0 ≤ k 1 -1 ≤ t ≤ k 1 , we have Taking the superior limit on both sides of (4.18), note that and t → +∞ ⇒ k 1 → +∞, we have   Proof For any small ε > 0, there exist t 0 and a set Ω ε ⊂ Ω such that P(Ω ε ) > 1ε and xy k(A+x+By) ≤ εx for t ≥ t 0 and ω ∈ Ω ε . Then This shows that the distribution of the process x(t) converges weakly to the measure with density π(x).

Discussion
In this work, we consider a stochastic eutrophication-chemostat model with impulsive dredging and pulse inputting on environmental toxicant. The sufficient condition for the extinction of microorganisms is obtained. The sufficient condition for the investigated system with unique ergodic stationary distribution is also obtained by the Lyapunov functions method. The results of mathematical analysis and numerical analysis show that the stochastic noise, impulsive diffusion, and pulse input on environmental toxicant play important roles in the extinction and survival of the microorganisms. These results indicate the effective and reliable controlling strategy for water resource management with eutrophication.