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Eutrophication removal problems have captured the attention of biologists, mathematicians, and environmental scientists. Within this framework, an impulsive eutrophication controlling system is studied analytically and numerically. A key advantage of the eutrophication system is that it can be quite accurate to describe the interaction effect of some critical factors (fishermen catch and releasing small fry, etc.), which enables a systematic and logical procedure for fitting eutrophication mathematical system to real monitoring data and experiment data. Mathematical theoretical works have been pursuing the investigation of two threshold functions of some critical parameters under the condition of all species persistence, which can in turn provide a theoretical basis for the numerical simulation. Using numerical simulation works, we mainly focus on how to choose the best value of some critical parameters to ensure the sustainability of the eutrophication system so that the eutrophication removal process can be well developed with maximizing economic benefit. These results may be further extended to provide a basis for simulating the algal bloom in the laboratory and understanding the application of some impulsive controlling models about eutrophication removal problems.

The eutrophication of lakes and reservoirs has been widely and intensively studied for several decades, which is a degradation process originating from introduction of nutrients from agricultural runoff and untreated industrial and urban discharges [

In 1932, Bertalanffy firstly proposed a method of using mathematical model to study the biological system [

Zeya reservoir of Wenzhou is located in subtropical regions. Because of eutrophication, the nuisance algal blooms and the serious zooplankton aggregation frequently come forth, which caused clogging and blocking of the filtration system and resulted in millions of people without drinking water. Eutrophication removal is achieved by two major processes, physicochemical and biological treatment techniques; specifically biological eutrophication removal from drinking water in the lake and reservoir is usually considered to accomplish optimal and economic eutrophication treatment [

Speaking on eutrophication of Zeya reservoir, as the algae population grows to a certain biomass level, the dominant zooplankton species (^{6} cell/L), the dominant zooplankton (^{2} piece/L), and the filter-feeding fish (silver carp and bighead carp) (1 fish/Cubic meter) at time

The main purpose of this paper is to enhance a theoretical and constructive framework by making the attempt to deal with eutrophication controlling problem for some drinking reservoirs using an impulsive eutrophication controlling system. Meanwhile, the theory discussions on the global asymptotical stability and permanent would help to design some efficient parameter constraint representations, which can in turn provide a theoretical basis for the numerical simulation. Further, in the context of population growth dynamics, how to implement impulsive control strategy to prevent and control the algal bloom is studied by simulating the dynamics of the impulsive controlling eutrophication system.

Now we will pursue the investigation of the constraint expression of the release amount

Let

The solution of system (

System (

The following lemma is obvious.

Let

We will use an important comparison theorem on impulsive differential equation.

Suppose

Finally, we give some basic properties about the following subsystems of system (

Clearly

For a positive periodic solution

There exists a constant

Therefore, we obtain the complete expression for a periodic solution

Let

The local stability of periodic solution

We will put (

Therefore, we have

The stability of the periodic solution

If each of these eigenvalues is less than one in magnitude, then the periodic solution

According to Floquet theory of impulsive differential equation,

In the following, we will prove the global attraction. Choose a

Note that

Hence,

Next, we prove that

Then, for any

Let

Now we study the stability of the periodic solution, which explains the threshold expression of the release amount

The system (

Suppose

We will prove this in the following two steps. First, let

Easy to prove that

Therefore, there exists a

Then

Second, if

The first equation of the system (

For

Let

Using the theoretical analysis, we obtain the threshold expression of the release amount

The effective methods used for eutrophication removal, algal bloom control process, and associated public perception are a crucial consideration in developing effective systems of eutrophication removal management for Zeya reservoir; hence the sustainability analysis of the impulsive eutrophication controlling system must be investigated in detail. In order to study how to choose the value of the parameters

(a) The relationship diagram of the function

The dynamics of the system (

Considering the great uncertainty that still surrounds biological treatment mechanism, it is most prudent to interpret the simulation output on the basis of information and data obtained from observation and experiment. Simulation design offers tremendous flexibility to study the behavior dynamic of biological treatment process, so it is necessary to investigate that the release amount

Bifurcation diagram of the system (

In real ecosystem, the effect of some critical factors can be depicted by the properties and the dynamic behaviors of species [

Bifurcation diagram of the system (

The dynamics of the system (

Based on the foregoing analysis, it is successful for impulsive eutrophication controlling system to implement the eutrophication removal process. The impulsive eutrophication controlling system not only can promote all wanted species persistence, but also is convenient for the manager to maintain the normal development of biological treatment process. Moreover, simulation analysis provides an approximation of the real biological controlling system behaviors; hence it is very easy to obtain the best value of

The theoretical ecologists and applied mathematicians are now working on controlling the algal bloom in ecosystem, but the optimal way to eradicate the algae population to control the algal bloom is yet to know, but in the context of population growth dynamics, how to implement impulsive control strategy to prevent and control the algal bloom with the maximizing economic benefits is worthy of studying. In the paper, we take an approach to study how to better implement impulsive eutrophication controlling system to control the algal bloom and maintain the normal development of biological treatment process with maximizing economic benefits by using mathematical analysis and simulating the dynamics. On the basis of eutrophication ecology and differential equation, an impulsive eutrophication controlling system is studied analytically and numerically. A key advantage of the impulsive eutrophication controlling system is that it can be quite accurate to describe the interaction effect of some critical factors (fishermen catch and releasing small fry, etc.), which enables a systematic and logical procedure for fitting eutrophication mathematical system to real monitoring data and experiment data. mathematical theoretical works have been pursuing the investigation of two threshold functions

From the viewpoint of population dynamics, numerical simulation works have been pursuing the investigation of confirming the existence and feasibility of Theorems

This work was supported by the Zhejiang Provincial Natural Science Foundation of China (Grant no. LY13A010010), by the Key Program of Zhejiang Provincial Natural Science Foundation of China (Grant no. LZ12C03001), by the National Natural Science Foundation of China (Grant no. 31370381), by the National Key Basic Research Program of China (973 Program, Grant no. 2012CB426510) and by the Nonprofit Technology Research Projects of Zhejiang Province (Grant no. 2011C23119).