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Discrete Dynamics in Nature and Society
Volume 2011, Article ID 163541, 17 pages
http://dx.doi.org/10.1155/2011/163541
Research Article

Complex Behavior in a Fish Algae Consumption Model with Impulsive Control Strategy

1School of Mathematics and Information Science, Wenzhou University, Wenzhou, Zhejiang 325035, China
2School of Life and Environmental Science, Wenzhou University, Wenzhou, Zhejiang 325035, China

Received 9 August 2011; Accepted 27 August 2011

Academic Editor: M. De la Sen

Copyright © 2011 Jin Yang and Min Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper investigates a dynamic mathematical model of fish algae consumption with an impulsive control strategy analytically. It is proved that the system has a globally asymptotically stable algae-eradication periodic solution and is permanent by using the theory of impulsive equations and small-amplitude perturbation techniques. Numerical results for impulsive perturbations demonstrate the rich dynamic behavior of the system. Further, we have also compared biological control with chemical control. All these results may be useful in controlling eutrophication.

1. Introduction

Controlling algae (in particular the deterioration of water caused by algae) has become an increasingly complex issue over the past two decades because economic loss will be enormous once the population of algae is out of control. At present, many of our lakes and large areas of algae bloom outbreaks per year [1, 2]; in these lakes, ecological balance is broken, the water quality is deteriorated, and human health is threatened. So research on how to control the population of algae is of great important theoretic and practical significance. Many methods have been used to control algal blooms.

Biological control is the practice of using natural enemies such as predators to suppress a prey population, as has already been done for pest control [3, 4]. In addition to the classical biological control based on predator-prey interaction, recently another form of biological control based on fish-algae interaction is extensively used. Many reservoirs have used the biological control methods to control algal blooms (they control algal blooms by stocking fish in the reservoir to graze algae directly), which has been proved to be effective in preventing the outbreak of algal blooms in East Lake in Wu Han province. However, many researchers doubt that this method is not only costly, but also cannot be effective in a few days. Another commonly used method is chemical control (usually dilution of copper sulfate), and this method can quickly kill a significant portion of the algae population, but it brings many negative impacts. Wherever possible, different methods should work together rather than against each other. In some cases, this can lead to synergy where the combined effect of different methods is greater than would be expected from simply adding the individual effects together [5]. Therefore, if we wish to eradicate the algae population, we should implement an impulsive control strategy which includes chemical control and biological control.

With the advance of the theory of impulsive differential equations [6, 7], impulsive differential equations are used to describe the evolving process and the control process of species [811], which make the models more reasonable [1214]. Moreover, the theory of impulsive differential equations is being recognized not only to be richer than the corresponding theory of differential equations without impulses, but also to represent a more natural framework for the mathematical modeling of real-world phenomenon [15, 16]. In this paper, we construct a mathematical model combining the fact of period biological control with chemical control; we first introduce a proportion periodic impulsive harvesting (fish) and chemical poisoning for the algae at time 𝑡=(𝑛+𝐿1)𝑇, and then we introduce a constant periodic releasing for natural enemies (fish) at time𝑡=𝑛𝑇; the system can be described as follows:𝑑𝑥(𝑡)𝑑𝑡=𝑢𝑐𝑥(𝑡)1𝑥(𝑡)/𝑥𝑚1𝑥(𝑡)/𝑥𝑐1𝑥2𝑢(𝑡)1𝑥(𝑡)𝑦(𝑡)𝑥(𝑡)+𝑘1,𝑑𝑦(𝑡)𝑑𝑡=𝑢3𝑦𝑢(𝑡)+2𝑥(𝑡)𝑦(𝑡)𝑥(𝑡)+𝑘1,𝑡𝑛𝑇,𝑡(𝑛+𝐿1)𝑇,Δ𝑥(𝑡)=𝛿1𝑥(𝑡),Δ𝑦(𝑡)=𝛿2𝑦(𝑡),𝑡=(𝑛+𝐿1)𝑇,Δ𝑥(𝑡)=0,Δ𝑦(𝑡)=𝑝,𝑡=𝑛𝑇,(1.1) where 𝑥(𝑡), 𝑦(𝑡) are the densities of the algae and fish at time 𝑡, Δ𝑥(𝑡)=𝑥(𝑡+)𝑥(𝑡), and Δ𝑦(𝑡)=𝑦(𝑡+)𝑦(𝑡); 𝑑𝑥(𝑡)/𝑑𝑡=𝑢𝑐𝑥(𝑡)((1(𝑥(𝑡)/𝑥𝑚))/(1(𝑥(𝑡)/𝑥))) is a mathematical model for a single population [17] and is established by Cui and Lawson; 𝑢𝑐 is a growth parameter which is related to the biological characteristics of populations and the rationalization of environmental resources; 𝑥𝑚(0𝑥𝑚/𝑥1) is the maximum density of the algae population (i.e., environmental carrying capacity); 𝑥 is a nutritional parameter which is related to the resource conditions of the environment; 𝑢1𝑥(𝑡)/(𝑥(𝑡)+𝑘1) is one of the most well-known functional responses describing a prey-predator interaction, called Holling-Type II functional response; 𝑐1 is the intraspecific competition rate of the algae; 𝑢3 is the average mortality rate for fish; 0𝛿1, 𝛿21 represent the fraction of the algae and fish which die due to the harvesting or chemical poisoning at𝑡=(𝑛+𝐿1)𝑇; 𝑝>0 is the number of fish released at time 𝑡=𝑛𝑇;𝑇 is the period of the impulsive effect; 𝑛 is the set of all nonnegative integers.

With model (1.1), we can take into account the effects in the external which can rapidly change the population densities. Impulsive reduction of the algae population density is possible after its partial destruction by poisoning with chemicals, and also impulsive increase of the fish population density is possible by artificial breeding or releasing the fish population; therefore, we can use impulsive control strategy to eradicate the algae population.

2. Preliminaries and Mathematical Analysis

Let 𝑅+=[0,),𝑅2+={𝑋𝑅2𝑋>0}. Denote that 𝑓=(𝑓1,𝑓2) is the map defined by the right-hand sides of the first and second equations of system (1.1). Let 𝑉𝑅+×𝑅2+𝑅+, then 𝑉 is said to belong to class 𝑉0 if(1)𝑉 is continuous in ((𝑛1)𝑇,(𝑛+𝑙1)𝑇]×𝑅2+, ((𝑛+𝑙1)𝑇,𝑛𝑇]×𝑅2+, and for each 𝑋𝑅2+,𝑛𝑁,lim(𝑡,𝑦)((𝑛+𝑙1)𝑇+,𝑋)𝑉(𝑡,𝑦)=𝑉((𝑛+𝑙1)𝑇+,𝑋) and lim(𝑡,𝑦)(𝑛𝑇+,𝑋)𝑉(𝑡,𝑦)=𝑉(𝑛𝑇+,𝑋) exist;(2)𝑉 is locally Lipschitzian in 𝑋.

Definition 2.1. Let 𝑉𝑉0; for (𝑡,𝑥)((𝑛1)𝑇,(𝑛+𝑙1)𝑇]×𝑅2+ and ((𝑛+𝑙1)𝑇,𝑛𝑇]×𝑅2+, the upper right derivative of 𝑉(𝑡,𝑋) with respect to the impulsive differential system (1.1) is defined as 𝐷+𝑉(𝑡,𝑋)=lim0+1sup[]𝑉(𝑡+,𝑋+𝑓(𝑡,𝑋))𝑉(𝑡,𝑋).(2.1)

Remark 2.2. (1) The solution of system (1.1) is a piecewise continuous function with 𝑋𝑅+𝑅2+, then 𝑋(𝑡) is continuous on ((𝑛1)𝑇,(𝑛+𝑙1)𝑇), and ((𝑛+𝑙1)𝑇,𝑛𝑇). (2) The smoothness properties of 𝑓 guarantee the global existence and uniqueness of solution of system (1.1) (for details, see book [6, 7]).

Lemma 2.3. Assume that 𝑋(𝑡) is a solution of system (1.1) such that(1)if 𝑋(0+)0, then 𝑋(𝑡)0 for all 𝑡0,(2)if 𝑋(0+)>0, then 𝑋(𝑡)>0, for all 𝑡>0.

Lemma 2.4. There exists a positive constant 𝑀>0 such that 𝑥(𝑡)𝑀 and 𝑦(𝑡)𝑀 for each solution of system (1.1) with all 𝑡 large enough.

If the algae population is eradicated, then system (1.1) will reduce to the following system: 𝑑𝑦(𝑡)𝑑𝑡=𝑢3𝑦𝑡𝑦(𝑡),𝑡(𝑛+𝑙1)𝑇,𝑡𝑛𝑇,+=1𝛿2𝑦𝑡𝑦(𝑡),𝑡=(𝑛+𝑙1)𝑇,+𝑦0=𝑦(𝑡)+𝑝,𝑡=𝑛𝑇,+=𝑦0.(2.2)

System (2.2) is a periodically forced linear system, then we get that𝑦(𝑡)=𝑝exp𝑢3(𝑡(𝑛1)𝑇)11𝛿2exp𝑢3𝑇𝑝,(𝑛1)𝑇<𝑡(𝑛+𝑙1)𝑇,1𝛿2exp𝑢3(𝑡(𝑛1)𝑇)11𝛿2exp𝑢3𝑇,(𝑛+𝑙1)𝑇<𝑡𝑛𝑇,(2.3) is a positive periodic solution of system (2.2) with the initial values 𝑦0+=𝑦𝑛𝑇+=𝑝11𝛿2exp𝑢3𝑇,𝑦𝑙𝑇+=𝑝1𝛿2exp𝑢3𝑙𝑇11𝛿2exp𝑢3𝑇,(2.4) since the general solution of (2.2) is 𝑦(𝑡)=1𝛿2𝑛1𝑦0+𝑝11𝛿2exp𝑢3𝑇exp𝑢3𝑇+𝑦(𝑡),(𝑛1)𝑇<𝑡(𝑛+𝑙1)𝑇,1𝛿2𝑛𝑦0+𝑝11𝛿2exp𝑢3𝑇exp𝑢3𝑇+𝑦((𝑡),𝑛+𝑙1)𝑇<𝑡𝑛𝑇.(2.5)

Then the following results can be got easily.

Lemma 2.5. 𝑦(𝑡) is a positive periodic solution of system (2.2),  and for every solution 𝑦(𝑡) of system (2.2), one has 𝑦(𝑡)𝑦(𝑡) as 𝑡.

Therefore, system (2.2) has an algae-eradication periodic solution (0,𝑦(𝑡)).

After the preliminaries, it is necessary to give the main theorems of this paper. Now, the conditions which assure the globally asymptotical stability of the an lgae-eradication periodic solution (0,𝑦(𝑡)) are given.

Theorem 2.6. If 𝑢𝑐𝑢𝑇1𝑝1𝛿2exp𝑢3𝑙𝑇1𝛿2exp𝑢3𝑇𝑢3𝑘111𝛿2exp𝑢3𝑇1<ln1𝛿1,(2.6) then the algae-eradication periodic solution (0,𝑦(𝑡)) is said to be globally asymptotically stable.

Proof. The local stability of the periodic solution (0,𝑦(𝑡)) may be determined by considering the behavior of small-amplitude perturbations of the solution. Define 𝑥(𝑡)=𝑢(𝑡),𝑦(𝑡)=𝑣(𝑡)+𝑦(𝑡), then the Linearization of system (1.1) becomes 𝑑𝑢(𝑡)=𝑢𝑑𝑡𝑐𝑢1𝑦(𝑡)𝑘1𝑢(𝑡),𝑑𝑣(𝑡)𝑑𝑡=𝑢3𝑢𝑣(𝑡)+2𝑢(𝑡)𝑦(𝑡)𝑘1,𝑡𝑛𝑇,𝑡(𝑛+𝐿1)𝑇,Δ𝑢(𝑡)=𝛿1𝑢(𝑡),Δ𝑣(𝑡)=𝛿2𝑣(𝑡),𝑡=(𝑛+𝐿1)𝑇,Δ𝑢(𝑡)=0,Δ𝑣(𝑡)=0,𝑡=𝑛𝑇,(2.7) and as a result, 𝑢(𝑡)𝑣(𝑡)=Φ(𝑡)𝑢(0)𝑣(0),0𝑡<𝑇,(2.8) where Φ(𝑡) satisfies 𝑑Φ(𝑡)=𝑢𝑑𝑡𝑐𝑢1𝑦(𝑡)𝑘10𝑢2𝑦(𝑡)𝑘1𝑢3Φ(𝑡),(2.9) and Φ(0)=𝐼, the identity matrix. The linearization of the third and fourth equations of (2.2) becomes 𝑢(𝑛+𝑙1)𝑇+𝑣(𝑛+𝑙1)𝑇+=1𝛿1001𝛿2𝑢((𝑛+𝑙1)𝑇)𝑣((𝑛+𝑙1)𝑇).(2.10) The linearization of fifth and sixth equations of (2.2) becomes 𝑢𝑛𝑇+𝑣𝑛𝑇+=1001𝑢(𝑛𝑇)𝑣(𝑛𝑇).(2.11) The stability of the periodic solution (0,𝑦(𝑡)) is determined by the eigenvalues of 𝜃=1𝛿1001𝛿21001Φ(𝑡).(2.12) Therefore, all eigenvalues of 𝜃 are given by 𝜆1=1𝛿1exp𝑇0𝑢𝑐𝑢1𝑦(𝑡)𝑘1𝑑𝑡,𝜆2=1𝛿2exp𝑢3𝑇<1.(2.13) According to Floquet theory, (0,𝑦(𝑡)) is locally asymptotically stable if 𝜆1<1, that is to say, 𝑢𝑐𝑢𝑇1𝑝1𝛿2exp𝑢3𝑙𝑇1𝛿2exp𝑢3𝑇𝑢3𝑘111𝛿2exp𝑢3𝑇1<ln1𝛿1.(2.14) In the following, we prove the global attractivity. Choose a 𝜀>0 such that 𝜉11𝛿1exp𝑇0𝑢𝑐𝑢1𝑘1𝑦(𝑡)𝜀𝑑𝑡<1,(2.15) and note that 𝑑𝑦(𝑡)/𝑑𝑡𝑢3𝑦(𝑡); from Lemma 2.5 and comparison theorem of impulsive equation, we get 𝑦(𝑡)>𝑦(𝑡)𝜀,(2.16) for all sufficiently large 𝑡. For simplification, assuming (2.16) holds for all 𝑡0. From (1.1) and (2.16), 𝑑𝑥(𝑡)𝑢𝑑𝑡𝑥(𝑡)𝑐𝑢1𝑘1𝑦𝑥𝑡(𝑡)𝜀,𝑡(𝑛+𝑙1)𝑇,+=1𝛿1𝑥(𝑡),𝑡=(𝑛+𝑙1)𝑇,(2.17) which leads to 𝑥((𝑛+𝑙)𝑇)𝑥(𝑛+𝑙1)𝑇+exp(𝑛+𝑙)𝑇(𝑛+𝑙1)𝑇𝑢𝑐𝑢1𝑘1𝑦(𝑡)𝜀𝑑𝑡=𝑥((𝑛+𝑙1)𝑇)1𝛿1exp(𝑛+𝑙)𝑇(𝑛+𝑙1)𝑇𝑢𝑐𝑢1𝑘1𝑦(𝑡)𝜀𝑑𝑡=𝑥((𝑛+𝑙1)𝑇)𝜉1.(2.18) Hence, 𝑥((𝑛+𝑙)𝑇)𝑥(𝑙𝑇)𝜉𝑛1 and 𝑥((𝑛+𝑙)𝑇)0 as 𝑛. Therefore, 𝑥(𝑡)0 when 𝑛, because 0<𝑥(𝑡)<𝑥((𝑛+𝑙1)𝑇)(1𝛿1)exp(𝑢𝑐𝑇) for (𝑛+𝑙1)𝑇<𝑡(𝑛+𝑙)𝑇.
Next, we prove that 𝑦(𝑡)𝑦(𝑡) as 𝑡. For 0<𝜀<(𝑢3𝑘1/𝑢2), there must exist a 𝑇>0 such that 0<𝑥(𝑡)<𝜀,𝑡𝑇. Without any loss of generality, we assume that 0<𝑥(𝑡)<𝜀 for all 𝑡0, then from system (1.1),𝑢3𝑦(𝑡)𝑑𝑦(𝑡)𝑑𝑡𝑢3+𝑢3𝑘1𝑢2𝜀𝑦(𝑡).(2.19) From Lemma 2.5 and comparison theorem of impulsive equation, 𝑧1(𝑡)𝑦(𝑡)𝑧2(𝑡) and 𝑧1(𝑡)𝑦(𝑡), 𝑧2(𝑡)𝑦(𝑡) as 𝑡, where 𝑧1(𝑡) and 𝑧2(𝑡) are solutions of 𝑑𝑧1(𝑡)𝑑𝑡=𝑢3𝑧1𝑧(𝑡),𝑡(𝑛+𝑙1)𝑇,𝑡𝑛𝑇,1𝑡+=1𝛿2𝑧1𝑧(𝑡),𝑡=(𝑛+𝑙1)𝑇,1𝑡+=𝑧1𝑧(𝑡)+𝑝,𝑡=𝑛𝑇,10+0=𝑦+,(2.20)𝑑𝑧2(𝑡)=𝑑𝑡𝑢3+𝑢3𝑘1𝑢2𝜀𝑧2𝑧(𝑡),𝑡(𝑛+𝑙1)𝑇,𝑡𝑛𝑇,2𝑡+=1𝛿2𝑧2𝑧(𝑡),𝑡=(𝑛+𝑙1)𝑇,2𝑡+=𝑧2𝑧(𝑡)+𝑝,𝑡=𝑛𝑇,20+0=𝑦+,(2.21) respectively, 𝑧2(𝑡)=𝑝exp𝑢3+𝑢3𝑘1/𝑢2𝜀(𝑡(𝑛1)𝑇)11𝛿2exp𝑢3+𝑢3𝑘1/𝑢2𝜀𝑇,𝑝(𝑛1)𝑇<𝑡(𝑛+𝑙1)𝑇,1𝛿2exp𝑢3+𝑢3𝑘1/𝑢2𝜀(𝑡(𝑛1)𝑇)11𝛿2exp𝑢3+𝑢3𝑘1/𝑢2𝜀𝑇,(𝑛+𝑙1)𝑇<𝑡𝑛𝑇.(2.22)
Therefore, for any 𝜀1>0, there exists a 𝑇1>0 such that𝑧1(𝑡)𝜀1𝑦(𝑡)𝑧2(𝑡)+𝜀1,for𝑡>𝑇1.(2.23) Let 𝜀0 such that 𝑦(𝑡)𝜀1𝑦(𝑡)𝑦(𝑡)+𝜀1,(2.24) for 𝑡 large enough, which implies 𝑦(𝑡)𝑦(𝑡) as 𝑡. This completes the proof.

Now, we investigate the permanence of system (1.1).

Theorem 2.7. System (1.1) is permanent provided 𝑢𝑐𝑢𝑇1𝑝1𝛿2exp𝑢3𝑙𝑇1𝛿2exp𝑢3𝑇𝑢3𝑘111𝛿2exp𝑢3𝑇1>ln1𝛿1(2.25) holds true.

Proof. Let 𝑋(𝑡)=(𝑥(𝑡),𝑦(𝑡)) be any solution of system (1.1) with 𝑋(0)>0. From Lemma 2.4, there exists a positive constant 𝑀 such that 𝑥(𝑡)𝑀 and 𝑦(𝑡)𝑀 for 𝑡 large enough. From (2.16), we have 𝑦(𝑡)>𝑦(𝑡)𝜀 for all sufficiently large 𝑡 and some 𝜀 such that 𝑦(𝑡)𝑝(1𝛿2)exp(𝑢3𝑇)/(1(1𝛿2)exp(𝑢3𝑇))𝜀𝜁2 for 𝑡 large enough. Therefore, it is only necessary to find an 𝜁1>0 such that 𝑥(𝑡)𝜁1 for 𝑡 large enough. We prove this in the following two steps.Step 1. Let 0<𝜁3<𝑢3𝑘1/𝑢2, 𝜀1>0 be small enough such that 𝜓𝑢𝑐𝜁13𝑥𝑚𝑇𝑐1𝜁3𝑢𝑇1𝜀1𝑘1𝑇𝑢1𝑘1𝑢1𝑝1𝛿2exp𝑢3+𝑢3𝑘1/𝑢2𝜁3𝑙𝑇1𝛿2exp𝑢3+𝑢3𝑘1/𝑢2𝜁3𝑇𝑢3𝑢3𝑘1/𝑢2𝜁3𝑘111𝛿2exp𝑢3+𝑢3𝑘1/𝑢2𝜁3𝑇>1,(2.26) then it is easy to prove that 𝑥(𝑡)<𝜁3 cannot hold for all 𝑡. Otherwise, 𝑑𝑦(𝑡)𝑑𝑡𝑢3+𝑢3𝑘1𝑢2𝜁3𝑦𝑡𝑦(𝑡),𝑡(𝑛+𝑙1)𝑇,𝑡𝑛𝑇,+=1𝛿2𝑦𝑡𝑦(𝑡),𝑡=(𝑛+𝑙1)𝑇,+𝑦0=𝑦(𝑡)+𝑝,𝑡=𝑛𝑇,+=𝑦0.(2.27) Then, 𝑦(𝑡)𝑧(𝑡) and 𝑧(𝑡)𝑧(𝑡)(𝑡), where 𝑧(𝑡) is the solution of 𝑑𝑧(𝑡)=𝑑𝑡𝑢3+𝑢3𝑘1𝑢2𝜁3𝑧𝑡𝑧(𝑡),𝑡(𝑛+𝑙1)𝑇,𝑡𝑛𝑇,+=1𝛿2𝑧𝑡𝑧(𝑡),𝑡=(𝑛+𝑙1)𝑇,+𝑧0=𝑧(𝑡)+𝑝,𝑡=𝑛𝑇,+0=𝑦+,𝑧(2.28)(𝑡)=𝑝exp𝑢3+𝑢3𝑘1/𝑢2𝜀(𝑡(𝑛1)𝑇)11𝛿2exp𝑢3+𝑢3𝑘1/𝑢2𝜀𝑇,𝑝(𝑛1)𝑇<𝑡(𝑛+𝑙1)𝑇,1𝛿2exp𝑢3+𝑢3𝑘1/𝑢2𝜀(𝑡(𝑛1)𝑇)11𝛿2exp𝑢3+𝑢3𝑘1/𝑢2𝜀𝑇,(𝑛+𝑙1)𝑇<𝑡𝑛𝑇.(2.29)
Therefore, there exists a 𝑇1>0 such that𝑦(𝑡)𝑧(𝑡)𝑧(𝑡)+𝜀1,(2.30) and it follows that 𝑑𝑥(𝑡)𝑢𝑑𝑡𝑥(𝑡)𝑐𝜁13𝑥𝑚𝑐1𝜁3𝑢1𝑘1𝑧(𝑡)+𝜀1𝑥𝑡,𝑡(𝑛+𝑙1)𝑇,+=1𝛿1𝑥(𝑡),𝑡=(𝑛+𝑙1)𝑇,(2.31) for 𝑡𝑇1. Let (𝑁+𝑙1)𝑇𝑇1, integrating (2.31) on ((𝑛+𝑙1)𝑇,(𝑛+𝑙)𝑇],𝑛𝑁, so 𝑥((𝑛+𝑙)𝑇)𝑥((𝑛+𝑙1)𝑇)1𝛿1exp(𝑛+𝑙)𝑇(𝑛+𝑙1)𝑇𝑢𝑐𝜁13𝑥𝑚𝑐1𝜁3𝑢1𝑘1𝑧(𝑡)+𝜀1𝑑𝑡=𝑥((𝑛+𝑙1)𝑇)𝜓,(2.32) then 𝑥((𝑁+𝑛+𝑙)𝑇)𝑥((𝑁+𝑙)𝑇)𝜓𝑛 when 𝑛; it is a contradiction because 𝑥(𝑡) is ultimately bounded. Therefore, there exists a 𝑡1>0 such that 𝑥(𝑡1)𝜁3.
Step 2. If 𝑥(𝑡)𝜁3 for all 𝑡>𝑡1, then the proof will be complete. Otherwise, let 𝑡=inf𝑡>𝑡1{𝑥(𝑡)<𝜁3}, then there are two possible cases for 𝑡.Case 1. If 𝑡=(𝑛1+𝑙1)T,𝑛1𝑁, then 𝑥(𝑡)𝜁3 for 𝑡[𝑡1,𝑡] and (1𝛿1)𝜁3𝑥(𝑡+)=(1𝛿1)𝑥(𝑡)<𝜁3, and select 𝑛2,𝑛3𝑁 such that 𝑛2𝜀1𝑇>ln1/𝑀+𝑝𝑢3+𝑢3𝑘1/𝑢2𝜁3,1𝛿1𝑛2𝑛exp2𝜓1𝑇𝜓𝑛3>1𝛿1𝑛2𝑛exp2𝜓+11𝑇𝜓𝑛31>1,(2.33) where 𝜓1=𝑢𝑐(1(𝜁3/𝑥𝑚))𝑐1𝜁3(𝑢1/𝑘1)𝑀<0. Let 𝑇=𝑛2𝑇+𝑛3𝑇; it is claimed here that there must be a 𝑡2(𝑡,𝑡+𝑇] such that 𝑥(𝑡2)>𝜁3. Otherwise, considering (2.28) with 𝑧(𝑡+)=𝑦(𝑡+), it follows that 𝑧(𝑡)=1𝛿2𝑛(𝑛1+1)𝑧𝑛1𝑇+𝑝11𝛿2exp𝑢3+𝑢3𝑘1/𝑢2𝜁3𝑇×exp𝑢3+𝑢3𝑘1𝑢2𝜁3𝑡𝑛1𝑇+𝑧(𝑡),(𝑛1)𝑇<𝑡(𝑛+𝑙1)𝑇,1𝛿2𝑛𝑛1𝑧𝑛1𝑇+𝑝11𝛿2exp𝑢3+𝑢3𝑘1/𝑢2𝜁3𝑇×exp𝑢3+𝑢3𝑘1𝑢2𝜁3𝑡𝑛1𝑇+𝑧(𝑡),(𝑛+𝑙1)𝑇<𝑡𝑛𝑇,(2.34) and 𝑛1+1𝑛𝑛1+𝑛2+𝑛3. Therefore, |𝑧(𝑡)𝑧(𝑡)|<(𝑀+𝑝)exp((𝑢3+(𝑢3𝑘1/𝑢2)𝜁3)(𝑡𝑛1𝑇))<𝜀1 and 𝑦(𝑡)𝑧(𝑡)𝑧(𝑡)+𝜀1 for 𝑛1𝑇+(𝑛21)𝑇𝑡𝑡+𝑇 which implies that (2.31) holds for 𝑡+𝑛2𝑇𝑡𝑡+𝑇. So as in Step 1, 𝑥𝑡𝑡+𝑇𝑥+𝑛2𝑇𝜓𝑛3.(2.35) From system (1.1) 𝑑𝑥(𝑡)𝑢𝑑𝑡𝑥(𝑡)𝑐𝜁13𝑥𝑚𝑐1𝜁3𝑢1𝑘1𝑀𝑥𝑡,𝑡(𝑛+𝑙1)𝑇,+=1𝛿1𝑥(𝑡),𝑡=(𝑛+𝑙1)𝑇,(2.36) for 𝑡[𝑡,𝑡+𝑛2𝑇]. Integrating (2.36) on 𝑡[𝑡,𝑡+𝑛2𝑇] such that 𝑥𝑡+𝑛2𝑇𝜁31𝛿1𝑛2𝑛exp2𝜓1𝑇,(2.37) thus, 𝑥(𝑡+𝑇)𝜁3(1𝛿1)𝑛2exp(𝑛2𝜓1𝑇)𝜓𝑛3>𝜁3, which is a contraction.
Let 𝑡2=inf𝑡>𝑡{𝑥(𝑡)>𝜁3}, then 𝑥(𝑡)𝜁3 when 𝑡(𝑡,𝑡2) and 𝑥(𝑡2)=𝜁3. For 𝑡(𝑡,𝑡2),𝑥(𝑡)𝜁31𝛿1𝑛2+𝑛3𝑛exp2+𝑛3𝜓1𝑇.(2.38) Let 𝜁1=𝜁3(1𝛿1)𝑛2+𝑛3exp((𝑛2+𝑛3)𝜓1𝑇), then we have 𝑥(𝑡)𝜁1 for 𝑡(𝑡,𝑡2). For 𝑡>𝑡2, the same arguments can be continued since 𝑥(𝑡2)𝜁3.
Case 2. If 𝑡(𝑛1+𝑙1)𝑇,𝑛1𝑁, then 𝑥(𝑡)𝜁3 for 𝑡[𝑡1,𝑡], and 𝑥(𝑡)=𝜁3; suppose that 𝑡((𝑛1+𝑙1)𝑇,(𝑛1+𝑙)𝑇),𝑛1𝑁. There are also two possible cases for 𝑡(𝑡,(𝑛1+𝑙)𝑇).Subcase 1. If 𝑥(𝑡)𝜁3 for all 𝑡(𝑡,(𝑛1+𝑙)𝑇), as in Case 1, we can prove that there must be a 𝑡1[(𝑛1+𝑙)𝑇,(𝑛1+𝑙)𝑇+𝑇] such that 𝑥(𝑡1)>𝜁3. Here, we omit it.
Let 𝑡3=inf𝑡>𝑡{𝑥(𝑡)>𝜁3}, then 𝑥(𝑡)𝜁3 when 𝑡(𝑡,𝑡3) and 𝑥(𝑡3)=𝜁3. For 𝑡(𝑡,𝑡3)𝑥(𝑡)𝜁31𝛿1𝑛2+𝑛3𝑛exp2+𝑛3𝜓+11𝑇.(2.39) Let 𝜁1=𝜁3(1𝛿1)𝑛2+𝑛3exp((𝑛2+𝑛3+1)𝜓1𝑇)<𝜁1, then 𝑥(𝑡)𝜁1 for 𝑡(𝑡,𝑡3), and when 𝑡>𝑡3, the same arguments can be got since 𝑥(𝑡3)𝜁3.
Subcase 2. There exists a 𝑡(𝑡,(𝑛1+𝑙)𝑇) such that 𝑥(𝑡)>𝜁3. Let 𝑡4=inf𝑡>𝑡{𝑥(𝑡)>𝜁3} such that 𝑥(𝑡)𝜁3 when 𝑡(𝑡,𝑡4) and 𝑥(𝑡4)=𝜁3. When 𝑡(𝑡,𝑡4), the inequality (2.36) holds. Integrating (2.36) on 𝑡(𝑡,𝑡4), then 𝑥𝑡(𝑡)𝑥𝜓exp1𝑡𝑡𝜁3𝜓exp1𝑇>𝜁1.(2.40) Since 𝑥(𝑡4)𝜁3 for 𝑡>𝑡4, the same arguments can be continued. Therefore, 𝑥(𝑡)>𝜁1 for 𝑡>𝑡1, so system (1.1) is permanent. The proof is complete.

3. Numerical Analysis

3.1. Bifurcation Analysis

The global dynamical behavior and the permanence of system (1.1) are investigated using numerical simulations; the following parameters and initial values were considered to substantiate our theoretical results: 𝑢1=0.175, 𝑢2=0.3, 𝑢3=0.18, 𝑢𝑐=0.5, 𝑥𝑚=15, 𝑥=20, 𝑘1=0.6, 𝑐1=0.05, 𝛿1=0.4, 𝛿2=0.4, 𝑝=4, 𝐿=0.07, 𝑥0=0.5, and 𝑦0=0.5.

From Theorem 2.6, it is known that algae-eradication periodic solution is globally asymptotically stable when 𝑇<𝑇max; this algae-eradication periodic solution (0,𝑦(𝑡)) is shown in Figure 1. It is clear that the variable predator 𝑦 oscillates in a stable cycle, but the algae 𝑥 rapidly decrease to zero, and 𝑇max9. If the period of the pulses 𝑇 is larger than 𝑇max, then the algae-eradication periodic solution becomes unstable, and it is possible that the algae and the fish population can coexist on a limit cycle when 𝑇>𝑇max (Figure 2), so system (1.1) can be permanent from Theorem 2.7. As the period of pulses increases, system (1.1) exhibits rich dynamic behaviors. In Figure 3, the typical bifurcation diagrams for system (1.1) were displayed with respect to 𝑇 in the range 𝑇[9,33]. When 9<𝑇<10.6, we can see 𝑇-period solution of system (1.1), and 𝑇-period solution is stable. When 𝑇>12.2, system (1.1) becomes unstable, and there is a cascade of period-doubling bifurcations leading to chaos (Figure 4). As 𝑇 further increases, the bifurcation diagrams show that system (1.1) exhibits rich dynamics including period-halving bifurcation, symmetry breaking pitchfork bifurcation, period-doubling bifurcation, quasiperiod oscillations, narrow or wide periodic windows, and crisis.

fig1
Figure 1: Dynamic behavior of system (1.1). When 𝑇<𝑇max9, the algae will be eradicated. Time series evolving according to biological control system (1.1) of (a) the algae population 𝑥, (b) the fish population 𝑦.
fig2
Figure 2: (a) A period 𝑇 attractor when 𝑇=10, (b) time series algae population when 𝑇=10.
fig3
Figure 3: Bifurcation diagrams for system (1.1) showing the effect of 𝑇 with 𝑝=4; we keep other parameters the same. (a) 𝑥 versus 𝑇, (b) 𝑦 versus 𝑇.
fig4
Figure 4: Strange attractor. (a) Chaotic attractor when 𝑇=11, (b) time series of algae population when 𝑇=11.

Then, we investigate the effect of the number of fish released 𝑝 to vary for system (1.1). Figure 5 shows the typical bifurcation diagrams of 𝑝 for 0<𝑝<9; it is clear that with the increasing number of fish released, system (1.1) shows complex behaviors including period-doubling bifurcations, chaotic band with wide or narrow periodic windows, crisis, tangent bifurcations, and period-halving bifurcation. When 𝑝>8.55, the algae will be eradicated, and the algae-eradication periodic solution occurs.

fig5
Figure 5: Bifurcation diagrams for system (1.1) showing the effect of 𝑝 with 𝑇=20; we keep other parameters the same. (a) 𝑥 versus 𝑝, (b) 𝑦 versus 𝑝.
3.2. The Largest Lyapunov Exponent

Convincing evidence for deterministic chaos has come from several recent experiments [18, 19]. From these results, the problem of detecting and quantifying chaos has become an important one; it is clear that chaos plays a very significant role in these studies [20, 21]. Therefore, the largest Lyapunov exponent is considered to be the most useful diagnostic tool for chaotic systems [2225]. The largest Lyapunov exponent 𝜆 must be positive for a chaotic attractor; otherwise, if 𝜆 is negative, the system will enter a stable state or become a periodic attractor. Reviewing the bifurcation diagram in Figures 3 and 5, we can calculate the corresponding largest Lyapunov exponent (𝑇 ranging from 9 to 23, 𝑝 ranging from 0 to 6.2) for system (1.1). The output is shown in Figure 6.

fig6
Figure 6: (a) The largest Lyapunov exponent (𝑇 ranging from 9 to 23) for system (1.1); (b) the largest Lyapunov exponent (𝑝 ranging from 0 to 6.2) for system (1.1).

4. Conclusions

In this paper, the effects of impulsive perturbations on a algae-fish consumption model have been investigated. The local and global stability of the algae-eradication periodic solution have been proved when the period of the pulses is less than critical values. In addition, conditions for the permanence of the system have been given by comparison theorem when the period of the pulse is larger than critical values. The largest Lyapunov exponent has been used to confirm the existence of chaotic dynamics.

From Theorem 2.6, the algae-eradication periodic solution is globally asymptotically stable when 𝑇<𝑇max. Therefore, in order to eradicate the algae population, we can take impulsive control strategy considering the effect caused by the chemical control to the environment and the cost of biological control when 𝑇<𝑇max. If we drive the fish population in a small pool or harvest the fish, then chemical poisoning will kill the algae population in large quantities, and the damage to fish population will be very small. If we only choose chemical control strategy (𝑝=0), from Theorem 2.6, the algae population and the fish population will be eradicated when 𝑇<(1/𝑢𝑐)ln(1/(1𝛿1))1.03; in this case, chemical control will not only destroy the biodiversity, but also cause damage to the environment, that is not desirable. If we only choose biological control strategy (𝛿1=0,𝛿2=0) and keep other parameters the same, then we have 𝑇<15.6; it is clear that biological control will cost much and take a long time to eradicate the algae population. Therefore, we should combine chemical control with biological control in order to control algal blooms efficiently.

Acknowledgments

The authors would like to thank the editor and the anonymous referees for their valuable comments and suggestions on this paper. This work was supported by the National Natural Science Foundation of China (Grant no. 30970305 and Grant no. 31170338) and also by the Zhejiang Provincial Natural Science Foundation of China (Grant no. Y505365).

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