# Analysis of Mathematics and Sustainability in an Impulsive Eutrophication Controlling System

Authors: Hengguo Yu; Min Zhao; Qi Wang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

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.


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## Body

## 1. Introduction

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 [1]. The eutrophication has been identified as a major and serious water quality management issue [2], not only in the developing countries [3–5], but also in the developed countries [6–8]. Impairment of water quality due to eutrophication can lead to a series of problems and result in losses of ecological integrity, sustainability, and safe use of aquatic ecosystems [9]. It is characterized by frequently recurring algal blooms, zooplankton aggregation, and reduction in species diversity in water bodies at all trophic levels [2].In 1932, Bertalanffy firstly proposed a method of using mathematical model to study the biological system [10]. He pointed out that this approach is a combination of coordination, order, and purpose, which in turn formed three basic ideas of studying the biological system, namely, trophic-level analysis, system perspective, and dynamic view. The main aim of modeling population dynamics is to improve the understanding of the interactions between populations and their dependence on internal and external conditions [11]. Hence, the mathematical models of biological population dynamics have to account not only for the growth and interactions between populations but also for the forecast of the disasters which can be caused by the plankton like blue-green algae, Moina and Brachionus. Early attempts began with the logistic growth, exponential growth, and Cui growth (this growth can explain the relationship between population increment and limiting resources) for biological population. With the increasing effects of the eutrophication, zooplankton aggregation, and algal blooms, the history of mathematical modeling plankton or zooplankton dynamics and biological eutrophication removal process is already quite long and has been initiated by biological science. These approaches have been refined to more realistic descriptions of the development of biological natural populations, especially algae population. In recent decades, the main researches focused on how to control the eutrophication, how to predict the outbreak of algae, how to simulate the spread tendency of algae, and how to develop eutrophication control strategy based on actual data. Hence, more and more researches have given an attempt to combine the forces of ecologist and mathematical engineers in an evaluation of possible control strategies for the management of eutrophication of water bodies, which is becoming an urgent water-management problem of drinking water reservoirs [12]. Malmaeus and Håkanson have depicted the development of a lake eutrophication model; they indicated that all changes that provided significant differences and improved predictions have been adopted [13]. Chen et al. said that their mathematical description provides a different insight into how to use nitrogen and phosphorus more effectively in the eutrophication control and elimination of blue-green blooms [14].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 [15]. In order to understand and improve the performance of biological eutrophication removal treatment, it is necessary to briefly look into the mechanisms of eutrophication removal. The removal process of eutrophication in the Zeya reservoir is complex and dynamic with many variables. Since successful management of eutrophication in Zeya reservoir is often dependent on proper identification of the most likely growth limiting amount for algae, special emphasis is given to the limiting amount concept. In order to apply the biological principle to control eutrophication in the Zeya reservoir, we give good biological control agents for algae. First, we bring up two species of filter-feeding fish, silver carp (Hypophthalmichthys molitrix), and bighead carp (Aristichthys nobilis) [16], which are thought to be suited to control algae and zooplankton biomass directly in freshwater reservoir, especially cyanobacteria. Second, because the decision-maker must give cost-benefit and maintenance considerations, fish should be carried out and controlled at an appropriate time. At the same time, in order to prevent the rapid growth of plankton, especially algae, and maintain a good circulating development of biological process, we must release a certain amount of filter-feeding fish at a fixed time. These processes are subject to short-term perturbations whose durations are too small compared to the duration of the process. Consequently, it is natural to assume that these perturbations act instantaneously in the form of impulses. Thus impulsive differential equations and differential equations involving impulsive effects appear as a natural mathematical description (model) of observed evolution phenomena of several real world problems [17–21]. These works have introduced the theory of impulsive differential equations and its applications and provided a theoretical basis of the application of impulsive differential equations. In recent years, Fan et al. [22] have proposed the dynamics of a class of mutualism-competition-predator interaction models with Beddington-DeAngelis functional responses and impulsive perturbations; this work is important for solving the global stability and the globally exponential stability of system. Yu and Lu [23] mainly focused on the permanence and global attractivity of a discrete almost periodic ratio-dependent Leslie system with time delays and feedback controls; these works vigorously promote the application of feedback control. Shen et al. [24] mainly have studied the permanence of a single species system with distributed time delay and feedback controls; it is a very important fact that the feedback control is harmless to the permanence of species; these works are significant for the application of feedback control in biological control. Furthermore, the main aim of this investigation is to answer that what is the fundamental mathematical controlling mechanism, which can support the biological principle to control eutrophication in the Zeya reservoir and to provide a different insight into how to make the effects of environmental factors in connection with the ability of controlling eutrophication.Speaking on eutrophication of Zeya reservoir, as the algae population grows to a certain biomass level, the dominant zooplankton species (Moina, Cyclops, and Brachionus are mainly grazing algae) are bound to sire and grow, so that these results can lead to damage or even crash of aquatic ecosystems. Furthermore, based on the above discussion, modeling is playing an increasing role in helping to identify causes and effective control measures through what if studies, and strategies are being identified for the management and control of eutrophication [25, 26]; the paper considers an eutrophication mathematical model, which can be described by the following differential equations:
(1)dx(t)dt=r1x(t)(1-x(t)k1)-a1x(t)y(t)b1+x(t)-a2x(t)z(t)b2+x(t),dy(t)dt=r2y(t)(1-y(t)k2)+e1a1x(t)y(t)b1+x(t)-a3y(t)z(t)b3+y(t),dz(t)dt=a2e2x(t)z(t)b2+x(t)+a3e3y(t)z(t)b3+y(t)-mz(t),e1a1x(t)y(t)b1+x(t)a3y(t)z(t)b3+y(t)t≠nT,t≠(n+l-1)T,Δx(t)=-δ1x(t),Δy(t)=-δ2y(t),Δz(t)=-δ3z(t),t=(n+l-1)T,Δx(t)=0,Δy(t)=0,Δz(t)=p,t=nT,
wherex(t),y(t), andz(t)are the biomass of the algae population (cyanobacteria) (106 cell/L), the dominant zooplankton (Moina, Cyclops, and Brachionus) (102 piece/L), and the filter-feeding fish (silver carp and bighead carp) (1 fish/Cubic meter) at timet, respectively,Δx(t)=x(t+)-x(t),Δy(t)=y(t+)-y(t), andΔz(t)=z(t+)-z(t).0<l<1is used to describe the intervals of time between the pulsed use of controls.ri(i=1,2) are the intrinsic growth rate,ai(i=1,2,3) andbi(i=1,2,3) measure the efficience of the prey in evading a predator attack.ei  (i=1,2,3) denote the efficiency with which resources are converted to new consumers,ki(i=1,2) are carrying capacity in the absence of the filter-feeding fish, andmis the mortality rates for filter-feeding fish. The decision-maker of biological treatment technology must implement an impulsive control strategy that can better maintain biological treatment process, so we not only release a certain amount of the filter-feeding fishrz(p>0is the release amount of the filter-feeding fishrz) att=nTto graze the algae and the dominant zooplankton, but also consider the algae and the dominant zooplankton losses due to other reasons as well as the filter-feeding fish harvest att=(n+l-1)T(δi(i=1,2,3) (0<δi<1) represent the fraction of three species), whereTis the period of the impulsive effect,n∈N, andNis the set of all nonnegative integers.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.


## 2. Mathematical Analysis

Now we will pursue the investigation of the constraint expression of the release amountpand the average harvestδ3of the filter-feeding fish. LetR+=[0,∞), andR+3={X∈R3∣X≥0}. Denotef=(f1,f2,f3)as the map defined by the right hand of the first, second, and third equations of system (1). LetV:R+×R+3→R+; thenVis said to belong to classV0if(1)
Vis continuous in(nT,(n+l-1)T)∪((n+l-1)T,(n+1)T]×R+3, and for eachX∈R+3,n∈N,lim(t,y)→(nT+,X)V(t,y)=V(nT+,X)exists;(2)
Vis locally Lipschitzian inX.Definition 1.
LetV∈V0; then for(t,x)∈(nT,(n+l-1)T)∪((n+l-1)T,(n+1)T]×R+3, the upper right derivative ofV(t,X)with respect to the impulsive differential system (1) is defined as
(2)D+V(t,X)=limh→0+sup1h[V(t+h,X+hf(t,x))-V(t,X)].
The solution of system (1) is a piecewise continuous functionX:R+→R+3,X(t)is continuous on(nT,(n+l-1)T)∪((n+l-1)T,(n+1)T],n∈N, andX(nT+)=limt→nT+X(t)exists. The smoothness properties offguarantee the global existence and uniqueness of solution of system (1); for details see [8, 9].Definition 2.
System (1) is said to be permanent if there exists a compactΩ⊂intR+3such that every solution(x(t),y(t),z(t))of system (1) will eventually enter and remain in the regionΩ.The following lemma is obvious.Lemma 3.
LetX(t)be a solution of system (1) withX(0+)≥0; thenX(t)≥0for allt≥0, and furtherX(t)>0,t>0ifX(0+)>0.We will use an important comparison theorem on impulsive differential equation.Lemma 4 (Bainov and Simeonov, 1993 [17]).
SupposeV∈V0. Assume that
(3)D+V(t,X)≤g(t,V(t,X)),t≠nT,t≠(n+l-1)T,V(t,X(t+))≤ψn1(V(t,X)),t=(n+l-1)T,V(t,X(t+))≤ψn2(V(t,X)),t=nT,
whereg:R+×R+2→R+2is continuous in((n-1)T,(n+l-1)T)and((n+l-1)T,nT]foru∈R+2,n∈N,lim(t,v)→(nT+,u)g(t,v)=g(nT+,u), andlim(t,v)→((n+l-1)T+,u)g(t,v)=g((n+l-1)T+,u)exists;ψni(i=1,2):R+→R+2is nondecreasing. Letr(t)be maximal solution of the scalar impulsive differential equation
(4)du(t)dt=g(t,u(t)),t≠nT,t≠(n+l-1)T,u(t+)=ψn1(u(t)),t=(n+l-1)T,u(t+)=ψn2(u(t)),t=nT,u(0+)=u0,
existing on[0,∞). ThenV(0+,X0)≤u0implies thatV(t,X(t))≤r(t),t≥0, whereX(t)is any solution of system (1).Finally, we give some basic properties about the following subsystems of system (1):
(5)dz(t)dt=-mz(t),t≠nT,t≠(n+l-1)T,z(t+)=(1-δ3)z(t),t=(n+l-1)T,z(t+)=z(t)+p,t=nT,(0+)=z0.Clearly(6)z*(t)={pexp(-m(t-(n-1)T))1-(1-δ3)exp(-mT)(n-1)T<t≤(n+l-1)T,p(1-δ3)exp(-m(t-(n-1)T))1-(1-δ3)exp(-mT)(n+l-1)T<t≤nT,n∈N,and
(7)z*(0+)=z*(nT+)=p1-(1-δ3)exp(-mT),z*(lT+)=p(1-δ3)exp(-mlT)1-(1-δ3)exp(-mT)
is a positive periodic solution of system (5). Since
(8)z(t)={(1-δ3)n-1(z(0+)-p1-(1-δ3)exp(-mT))×exp(-mt)+z*(t),(n-1)T<t≤(n+l-1)T,(1-δ3)n(z(0+)-p1-(1-δ3)exp(-mT))×exp(-mt)+z*(t),(n+l-1)T<t≤nT
is the solution of system (5) with initial valuez0=z(0+)≥0, wheret∈((n-1)T,(n+l-1)T)∪((n+l-1)T,nT],n∈N, we get the following.Lemma 5.
For a positive periodic solutionz*(t)of system (5) and every solutionz(t)of system (5) withz0≥0, one has|z(t)-z*(t)|→0,t→∞.Lemma 6.
There exists a constantMsuch thatx(t)≤M,y(t)≤M,z(t)≤M, andp(t)≤Mfor each solution(x(t),y(t),z(t),p(t))of system (1) with alltlargest enough.Therefore, we obtain the complete expression for a periodic solution(0,0,z*(t))of system (1). Now, we study the stability of the periodic solution, which explains the threshold expression of the release amountpand the average harvestδ3of the filter-feeding fish under the condition of some species extinction.Theorem 7.
Let(x(t),y(t),z(t))be any solution of system (1); then(0,0,z*(t))is said to be globally asymptotically stable if
(9)r1T-a2b2mp1-(1-δ3)exp(-mT)×(1-δ3exp(-mlT)-(1-δ3)exp(-mT))<ln(11-δ1),r2T-a3b3mp1-(1-δ3)exp(-mT)×(1-δ3exp(-mlT)-(1-δ3)exp(-mT))<ln(11-δ2).Proof.
The local stability of periodic solution(0,0,z*(t))may be determined by considering the behavior of small amplitude perturbation of the solution. Define
(10)x(t)=u(t),y(t)=v(t),z(t)=w(t)+z*(t).
We will put (10) into (1). The linearization of the system becomes
(11)du(t)dt=(r1-a2z*(t)b2)u(t),dv(t)dt=(r2-a3z*(t)b3)v(t)dw(t)dt=-mw(t),t≠nT,t≠(n+l-1)T,Δu(t)=-δ1u(t),Δv(t)=-δ2v(t),Δw(t)=-δ3w(t),t=(n+l-1)T,Δu(t)=0,Δv(t)=0,Δw(t)=0,t=nT.
Therefore, we have(12)(u(t)v(t)w(t))=Φ(t)(u(0)v(0)w(0)),0≤t<T,
whereΦ(t)satisfies
(13)dΦdt=(r1-a2z*(t)b2000r2-a3z*(t)b3000-m)Φ(t)
andΦ(0)=I, the identity matrix, and
(14)(u((n+l-1)T+)v((n+l-1)T+)w((n+l-1)T+))=(1-δ10001-δ20001-δ3)(u((n+l-1)T)v((n+l-1)T)w((n+l-1)T)),(u(nT+)v(nT+)w(nT+))=(100010001)(u(nT)v(nT)w(nT)).
The stability of the periodic solution(0,0,z*(t))is determined by the eigenvalues of
(15)Θ=(1-δ10001-δ20001-δ3)(100010001)Φ(T).
If each of these eigenvalues is less than one in magnitude, then the periodic solution(0,0,z*(t))is locally stable. Therefore all eigenvalues ofΘ, namely,
(16)u1=(1-δ1)exp(∫0T‍(r1-a2z*(t)b2)dt),u2=(1-δ2)exp(∫0T‍(r2-a3z*(t)b3)dt),u3=(1-δ3)exp(-mT)<1.
According to Floquet theory of impulsive differential equation,(0,0,z*(t))is locally asymptotically stable if|u1|<1and|u2|<1; that is to say,
(17)r1T-a2b2mp1-(1-δ3)exp(-mT)×(1-δ3exp(-mlT)-(1-δ3)exp(-mT))<ln(11-δ1),r2T-a3b3mp1-(1-δ3)exp(-mT)×(1-δ3exp(-mlT)-(1-δ3)exp(-mT))<ln(11-δ2).
In the following, we will prove the global attraction. Choose aɛ>0such that
(18)ξ1≜(1-δ1)exp(∫0T‍(r1-a2(z*(t)-ϵ)b2+M)dt)<1.
Note thatdz(t)/dt≥-mz(t); from Lemma 5 and comparison theorem of impulsive equation, we have
(19)z(t)>z*(t)-ϵ
for alltlarge enough. For simplification, we may assume that (19) holds for allt≥0. From (1) and (19), we can obtain
(20)dx(t)dt≤(r1-a2(z*(t)-ϵ)b2+M)x(t),t≠(n+l-1)T,x(t+)=(1-δ1)x(t),t=(n+l-1)T,
which leads to
(21)x((n+l)T)≤x((n+l-1)T+)×exp(∫(n+l-1)T(n+l)T‍(r1-a2(z*(t)-ϵ)b2+M)dt)=x((n+l-1)T)(1-δ1)×exp(∫(n+l-1)T(n+l)T‍(r1-a2(z*(t)-ϵ)b2+M)dt)=x((n+l-1)T)ξ1.
Hence,x((n+l)T)≤x(lT)ξ1nandx((n+l)T)→0asn→∞. Therefore,x(t)→0asn→∞since0<x(t)<x((n+l-1)T)(1-δ1)exp(r1T)for(n+l-1)T<t≤(n+l)T. Similarly,y(t)→0asn→∞.
Next, we prove thatz(t)→z*(t)ast→∞. For0<ϵsufficiently small, there must exist aT′>0such that0<x(t)<ϵand0<y(t)<ϵ,t≥T′. Without loss of generality, we may assume that0<x(t)<ϵand0<y(t)<ϵfor allt≥0; then from system (1) we obtain
(22)-mz(t)≤dz(t)dt≤(-m+γϵ)z(t),γ=e2a2b2+e3a3b3.
From Lemmas 4 and 5, we havev1(t)≤z(t)≤v2(t)andv1(t)→z*(t),v2(t)→v*(t)ast→∞, wherev1(t)andv2(t)are solutions of
(23)dv1(t)dt=-mv1(t),t≠nT,t≠(n+l-1)T,v1(t+)=(1-δ3)v1(t),t=(n+l-1)T,v1(t+)=v1(t)+p,t=nT,v1(0+)=z(0+),(24)dv2(t)dt=(-m+γϵ)v2(t),t≠nT,t≠(n+l-1)T,v2(t+)=(1-δ3)v2(t),t=(n+l-1)T,v2(t+)=v2(t)+p,t=nT,v2(0+)=z(0+),
respectively,
(25)v*(t)={pexp((-m+γϵ)(t-(n-1)T))1-(1-δ3)exp((-m+γϵ)T),(n-1)T<t≤(n+l-1)T,p(1-δ3)exp((-m+γϵ)(t-(n-1)T))1-(1-δ3)exp((-m+γϵ)T),(n+l-1)T<t≤nT.
Then, for anyϵ1>0there exists aT1>0such that
(26)z*(t)-ϵ1<z(t)<z*(t)+ϵ,t>T1.
Letϵ→0; we have
(27)z*(t)-ϵ1<z(t)<z*(t)+ϵ1
fortlarge enough, which impliesz(t)→z*(t)ast→∞. This completes the proof.Now we study the stability of the periodic solution, which explains the threshold expression of the release amountpand the average harvestδ3of the filter-feeding fish under the condition of all species persistence.Theorem 8.
The system (1) is permanent if
(28)r1T-a2b2mp1-(1-δ3)exp(-mT)×(1-δ3exp(-mlT)-(1-δ3)exp(-mT))>ln(11-δ1),(29)r2T-a3b3mp1-(1-δ3)exp(-mT)×(1-δ3exp(-mlT)-(1-δ3)exp(-mT))>ln(11-δ2).Proof.
SupposeX(t)=(x(t),y(t),z(t),p(t))is any solution of the system (1) withX(0)>0. From Lemma 6, we assume thatx(t)≤M,y(t)≤M, andz(t)≤Mwitht≥0. From (19), we havez(t)>z*(t)-ϵfor alltlarge enough and somez(t)≥p(1-δ3)exp(-mT)/(1-(1-δ3)exp(-mT))-ϵ≜ζ1fortlarge enough. Thus we only need to findζ2andζ8such thatx(t)>ζ2andy(t)>ζ8fortlarge enough. Now we will find aζ2.
We will prove this in the following two steps. First, let0<ζ4,ϵ1>0be small enough such that
(30)η1≜(1-δ1)exp(∫(n+l-1)T(n+l)T(-a2b2(v3*(t)+ϵ1))dtr1(1-ζ4k1)-a1Mb12222222222-a2b2(v3*(t)+ϵ1))dt)>1.
Easy to prove thatx(t)<ζ4cannot hold allt≥0. Otherwise,
(31)dz(t)dt≤-(m-e2a2x(t)b2-e3a3y(t)b3)z(t)≤-(m-e2a2ζ4b2-e3a3Mb3)z(t),NNNNt≠nT,t≠(n+l-1)T,z(t+)=(1-δ3)z(t),t=(n+l-1)T,z(t+)=z(t)+p,t=nT,z(0+)=z0.
Then we havez(t)≤v3(t)andv3(t)→v3*(t)(t→∞), wherev3(t)is the solution of
(32)dv3(t)dt=-(m-e2a2ζ4b2-e3a3Mb3)v3(t),22222222t≠nT,t≠(n+l-1)T,v3(t+)=(1-δ3)v3(t),t=(n+l-1)T,v3(t+)=v3(t)+p,t=nT,v3(0+)=z0,(33)
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(34)z(t)≤v3(t)<v3*(t)+ϵ1,(35)dx(t)dt≥(r1(1-ζ4k1)-a1Mb1-a2b2(v3*(t)+ϵ1))x(t),t≠nT,t≠(n+l-1)T,x(t+)=(1-δ1)x(t),t=(n+l-1)T
fort>T1. LetN1∈Nand(N1+l-1)T≥T1. Integrating (35) on((n+l-1)T,(n+l)T],n>N1, we can get
(36)x((n+l)T)≥x((n+l-1)T+)×exp(∫(n+l-1)T(n+l)T‍(-a2b2(v3*(t)+ϵ1))dtr1(1-ζ4k1)-a1Mb12222222222222222222-a2b2(v3*(t)+ϵ1))dt)=x((n+l-1)T)(1-δ1)×exp(∫(n+l-1)T(n+l)T‍(-a2b2(v3*(t)+ϵ1))dtr1(1-ζ4k1)-a1Mb12222222222222222222-a2b2(v3*(t)+ϵ1))dt)=x((n+l-1)T)η1.Thenx((N1+n+l))T≥X((N1+l)T)η1n→∞asn→∞, which is a contradiction to the boundedness ofx(t). Hence there exists at1>0such thatx(t1)≥ζ4. So, there exists at3>t1such thatx(t3)≥ζ4.Second, ifx(t)≥ζ4for allt≥t3, then our aim is obtained. Hence we only need to consider those solutions which leave the regionR={X(t)∈R+3:x(t)≤ζ2,y(t)≤ζ8,z(t)≤ζ1}and reenter it again. Lett*=inft≥t3{x(t)<ζ4}, there are two possible cases fort*.Case I. Considert*=(n1+l-1)T,n1∈N. Thenx(t)≥ζ4fort∈[t3,t*]and(1-δ1)ζ4≤x(t*+)=(1-δ1)x(t*)<ζ4. Selectn2,n3∈Nsuch that
(37)(n2-1)T>ln(ϵ1/(M+p))(-m+(e2a2ζ4/b2)+(e3a3M/b3)),(1-δ1)n2exp((n2η2T)η1n3)>(1-δ1)n2exp((n2+1)η2T)η1n3>1,
whereη2=(r1(1-(ζ4/k1))-(a1M/b1)-(a2M/b2))<0. LetT¯=n2T+n3T. We claim that there must be at4∈(t*,t*+T¯]such thatx(t)>ζ4. Otherwise, consider (32) withv3(t*+)=z(t*+), and we have(38)v3(t)={(1-δ3)n-(n1+1)(v3(n1T+)-p1-(1-δ3)exp((-m+(e2a2ζ4/b2)+(e3a3M/b3))T))×exp((-m+e2a2ζ4b2+e3a3Mb3)(t-n1T))+v3*(t),(n-1)T<t≤(n+l-1)T,(1-δ3)(n-1)(v3(n1T+)-p1-(1-δ3)exp((-m+(e2a2ζ4/b2)+(e3a3M/b3))T))×exp((-m+e2a2ζ4b2+e3a3Mb3)(t-n1T))+v3*(t),(n+l-1)T<t≤nTandn1+1≤n≤n1+n2+n3. Then
(39)|v3(t)-v3*(t)|<(M+p)exp((-m+e2a2ζ4b2+e3a3Mb3)(t-n1T))<ϵ1,z(t)≤v3(t)≤V3*(t)+ϵ1,(n1+n2+n3)T≤t≤t*+T¯,which implies that (35) holds fort*+n2T≤t≤t*+T¯. As in the first step, we have
(40)x(t*+n2T)≥x(t*+n2T)η1n3.The first equation of the system (1) gives
(41)dx(t)dt≥(r1(1-ζ4k)-a1Mb1-a2Mb2)x(t),t≠nT,t≠(n+l-1)T,x(t+)=(1-δ1)x(t),t=(n+l-1)T.
Integrating (41) on[t*,t*+n2T], we have
(42)x(t*+n2T)≥ζ4(1-δ1)n2exp(n2η2T).
Thusx(t*+T¯)≥ζ4(1-δ1)n2exp(n2η2T)η1n3>ζ4, and this is a contradiction. Lett¯=inft≥t*{x(t)>ζ4}, and thenx(t)≤ζ4,x(t¯)=ζ4, fort∈(t*,t¯); we have
(43)x(t)≥ζ4(1-δ1)n2+n3exp((n2+n3)η2T)≜ζ5.Fort>t¯, the same arguments can be continued sincex(t¯)≥ζ4.Case II. Considert*≠(n1+l-1)T,n1∈N. Thenx(t)≥ζ4fort∈[t3,t*]andx(t*)=ζ4, and supposet*∈((n1′+l-1)T,(n1′+l)T),n1′∈N.There are two possible cases fort∈(t*,(n1′+l)T).Case IIa.(x(t)≤ζ4for allt∈(t*,(n1′+l)T)).Similar to Case I, we can prove that there must be at2′∈[(n1′+l)T,(n1′+l)T+T¯]such thatx(t1′)>ζ4.Here we omit it. Lett~=inft≥t*{x(t)>ζ4}; thenx(t~)≤ζ4. Fort∈(t*,t~), we have
(44)x(t)≥ζ4(1-δ1)n2n3exp((n2+n3+1)η2T).Letζ6≜ζ4(1-δ1)n2n3exp((n2+n3+1)η2T)<ζ5, sox(t)≥ζ6fort∈(t*,t~).Fort>t~,the same arguments can be continued sincex(t~)≥ζ4.Case IIb. There exists at∈(t*,(n1′+l)T)such thatx(t)>ζ4.Lett~=inft≥t*{x(t)>ζ4}; thenx(t~)≤ζ4fort∈(t*,t~)andx(t~)=ζ4.Fort∈(t*,t~),(41) holds true. Integrating (41) on(t*,t~), we have
(45)x(t)≥x(t*)exp(η2(t-t*))≥ζ4exp(η2T)>ζ6.
Sincex(t)≥ζ4fort>t~,the same arguments can be continued. Hencex(t)≥ζ6for allt>t~.The proved method fory(t)is similar to the proved method forx(t). SetΩ={(x,y,x):x≥ζ2,y≥ζ8,z≥ζ1,x+y+z≤3M}.Obviously, we know that the setΩ∈intR+3is global attractors. every solution of system (1) will eventually enter and remain in regionΩ. Therefore, system (1) is permanent. The proof is completed.Using the theoretical analysis, we obtain the threshold expression of the release amountpand the average harvestδ3of the filter-feeding fish under the condition of some species extinction and all species persistence, which in turn provides a theoretical basis for the numerical simulation.


## 3. Numerical Analysis

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 parameterspandδ3so that they can satisfy the actual monitoring data and enable Theorems 7 and 8 to be feasible, it is necessary to present a more in-depth analysis of the relationship amongpandδ3. Setf1(p,δ3)=r1T-(a2/b2m)(p/(1-(1-δ3)exp(-mT)))(1-δ3exp(-mlT)-(1-δ3)exp(-mT))-ln(1/(1-δ1)), andf2(p,δ3)=r2T-(a3/b3m)(p/(1-(1-δ3)exp(-mT)))(1-δ3exp(-mlT)-(1-δ3)exp(-mT))-ln(1/(1-δ2)). On the basis of the actual monitoring data of the Zeya reservoir from year 2005 to year 2011 and laboratory controlling experiment data, the values of the parameters can be taken as follows:r1=0.75,r2=0.8,k1=20,k2=10,a1=0.3,a2=0.35,a3=0.5,e1=0.85,e2=0.5,e3=0.45,b1=2,b2=2.2,b3=2.5,m=0.15,δ1=0.15,δ2=0.25,l=0.003, andT=30. Hence we havef1(p,δ3)=-22.33748107+(1.060606061(0.9888910035-0.9754817198δ3)p)/(0.9888910035+0.01110899654δ3)andf2(p,δ3)=-23.71231793+(1.333333333(0.9888910035-0.9754817198δ3)p)/(0.9888910035+0.01110899654δ3). It is obvious to find from Figure 1(a) that there exist some values ofpandδ3so thatf1(p,δ3)<0andf2(p,δ3)<0orf1(p,δ3)>0andf2(p,δ3)>0, which in turn confirms the existence of Theorems 7 and 8. From the actual stocking amount of the little filter-feeding fish in Zeya reservoir to control the algal bloom, we can choose biologically feasible parametric numerical value ofpasp=35. Further, it can be seen from Figure 1(b) that if the value ofδ3is less than0.4, there existf1(p,δ3)>0andf2(p,δ3)>0, but if the value ofδ3is greater than or equal to 0.4, then the system (1) remains locally asymptotically stable. It is to be noted from Figure 2 that the time series solution of the filter-feeding fish population oscillates with stable limit cycle with increasing time. In contrast, both the algae population and the dominant zooplankton population rapidly decrease to their extinction with the increasing time. Meanwhile, these results further demonstrate that Theorems 7 and 8 are correct and feasible.(a) The relationship diagram of the functionf1(p,δ3)and the functionf2(p,δ3)with respect to the parameterspandδ3; (b) the relationship diagram of the functionf1(p,δ3)and the functionf2(p,δ3)with respect to the parameterδ3withp=35.
(a)
(b)The dynamics of the system (1).
(a)
(b)
(c)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 amountpis how to affect the dynamics of biological treatment process with the parameterδ3=0.4and0<p≤35. It is worth pointing out from Figure 3 that the system (1) with0<p<35exhibits relatively complex dynamical behaviors: chaos, period-halving bifurcation, and periodic attractor, but the system (1) withp=35remains locally asymptotically stable. Furthermore, it should be stressed that the dominant zooplankton population rapidly decreases to their extinction when the value ofpis greater than 4.4268; this phenomenon shows that the dominant zooplankton population will advance toward extinction with the increase ofp. But it is regretful for this phenomenon because the dominant zooplankton population is one of the main food for the filter-feeding fish and can graze the algae population, and the idea of using biological eutrophication removal treatment to control the algal bloom not only can control the algal bloom, but also consider the sustainable development of eutrophication removal process with the maximizing economic benefits. Thus, controlling the desirable biomass of the dominant zooplankton population is a primary aim of operation and maintenance, and how to well carry out the eutrophication removal process is an extremely vital problem of Zeya reservoir.Bifurcation diagram of the system (1) with0<p≤35.
(a)
(b)
(c)In real ecosystem, the effect of some critical factors can be depicted by the properties and the dynamic behaviors of species [27, 28]. To more carefully investigate how to legitimately implement eutrophication removal treatment, Figure 4 depicts the long-run dynamics of system (1) for0≤δ3≤1withp=5. It is easy to find from Figure 4 that the system exhibits relatively complex dynamical behaviors: chaos, period-halving bifurcation, and periodic attractor, but it is worth pointing out that, when the value ofδ3is beyond 0.6, the algae population will rapidly go to extinction and the dominant zooplankton population will go toward a reasonable state, and the biomass of the dominant zooplankton population is so small that it does not cause the aggregation of the dominant zooplankton. Meanwhile, the biomass of the filter-feeding fish is feasible to maintain the sustainable development of eutrophication removal process with the maximizing economic benefits because ofp=5. This result suggests that we can choose the value ofpandδ3as 5 and 0.6 for legitimately implementing eutrophication removal treatment. For further analysis of the feasibility of this result, Figure 5 gives the dynamics of the system (1) with the increase oftas well asp=5andδ3=0.6. It is gratifying to see from Figure 5 that the algae population quickly goes to the extinct status with the increase oft, and the dominant zooplankton population is close to a reasonable number with the increase oft. It is worth encouraging that the filter-feeding fish can arrive to a viable amount so that the eutrophication removal process can be developed with the maximizing economic benefits. Therefore, it should be stressed that these results can perfectly confirm that the impulsive eutrophication controlling system can be well applied in the process of controlling the algal bloom in the theoretical work of eutrophication removal and mathematical model.Bifurcation diagram of the system (1) with0≤δ3≤1.
(a)
(b)
(c)The dynamics of the system (1) withp=5andδ3=0.6.
(a)
(b)
(c)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 ofpandδ3, which is in accordance with observations and in turn shows that the selection of the best value about some critical parameters in the view of population dynamics is feasible.


## 4. Conclusions and Remarks

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 functionsf1(p,δ3)andf2(p,δ3)of two critical parameterspandδ3, iff1(p,δ3)<0andf2(p,δ3)<0, the system (1) remain locally asymptotically stable, and iff1(p,δ3)>0andf2(p,δ3)>0, the system (1) can ensure all species persistence. These results in turn provide a theoretical basis for the numerical simulation.From the viewpoint of population dynamics, numerical simulation works have been pursuing the investigation of confirming the existence and feasibility of Theorems7 and 8 and how to choose the best value of two critical parameterspandδ3. Within this framework, it can be assumed that the sustainability of the impulsive eutrophication controlling system mainly depends on two critical parameterspandδ3. Then the direct and indirect effects caused by critical parameterspandδ3are investigated by means of bifurcation analysis. These results indicate that ifp=5andδ3=0.6, the algae population will go to extinction, but the dominant zooplankton and the filter-feeding fish can survive in a feasible range, which can assure that the eutrophication removal process can be circulatingly developed with maximizing economic benefits. In addition, it is worthwhile to remark that mathematical theoretical results and numerical simulation results are complementary. It is hopeful that these results have an important implication for eutrophication removal management; hence in the follow-up works, we will use the system to character the growth rule and the correlation between the algae population, the dominant zooplankton, and the filter-feeding fish.



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*Source: 726172-2013-11-26.xml*