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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 452789, 17 pages
http://dx.doi.org/10.1155/2012/452789
Research Article

A Mathematical Model for the Dynamics of a Fish Algae Consumption Model with Impulsive Control Strategy

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

Received 21 October 2011; Accepted 7 February 2012

Academic Editor: Leevan Ling

Copyright © 2012 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

A dynamic mathematical model of fish algae consumption with an impulsive control strategy is proposed and analyzed in detail. It is shown that the system has a globally asymptotically stable algae-eradication periodic solution which can be obtained using the Floquet theory of impulsive differential equations and small-amplitude perturbation techniques. The conditions for the permanence of the system can also be determined. Numerical results for impulsive perturbations show the rich dynamic behavior of the system. All these results may be useful in controlling eutrophication.

1. Introduction

One of the most common ecological and environmental problems of inland water bodies is eutrophication, which diminishes water quality by spurring excessive growth of algae and increasing amounts of suspended organic material [13]. Because of climate change and discharges of industrial wastewater into rivers, the self-purification capacity of the water body decreases. This provides favorable conditions for algae growth, leading to repeated outbreaks of abnormal proliferations of algae blooms in tributaries; recently, this phenomenon has been increasing [4]. Therefore, research on how to control the amount of algae in the environment is of great theoretical importance and practical significance.

There are many ways to control algae populations. The most widely used method is chemical control; this is not discussed in detail here because it has many negative impacts. More recently, another important method, called biological control, has been introduced. Biological control is the practice of using natural enemies (fish) to suppress algae, as has already been done for pest control [58]. These natural enemies can prevent algae populations from increasing to levels that cause eutrophication.

Mathematical models of ecological population dynamics have not only to account for growth and interactions, but also to improve the understanding of the functioning of food chains and webs and their dependence on some environmental conditions [912]. Impulsive differential equations have been widely used to study the mathematical properties of impulsive predator-prey models or three-species food-chain models [1317]. 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 phenomena [1820].

Recently, because of eutrophication, the nuisance algal (cyanobacteria and green algae) blooms frequently come forth in the Zeya reservoir which is located in Wenzhou. The algal blooms can cause clogging and blocking of the filtration system and result in millions of people without drinking water. In order to apply the biological principle to control algal blooms in the Zeya reservoir, we carry out good biological control agents for the algal blooms. Firstly, we bring up two species of filter-feeding fish, silver carp (Hypophthalmichthys molitrix) and bighead carp (Aristichthys nobilis), which are thought to be suited to control algal biomass directly in freshwater reservoir, especially cyanobacteria and green algae. Secondly, the decision maker must give cost-benefit and maintenance considerations; the fish should be harvested at an appropriate time. Finally, in order to prevent the rapid growth of the cyanobacteria and green algae, we must release a certain amount of filter-feeding fish at a fixed time. These processes are subject to short-term perturbations. Consequently, it is obvious to assume that these perturbations act instantaneously in the form of impulses. Hence, a fish algae consumption model can be described by the following differential equations:𝑑𝑥(𝑡)𝑑𝑡=𝑟1𝐺𝑥(𝑡)0𝑥(𝑡)𝐺1𝑥(𝑡)𝑐1𝑥2𝑢(𝑡)1𝑥(𝑡)𝑧(𝑡)𝑏1,+𝑥(𝑡)𝑑𝑦(𝑡)𝑑𝑡=𝑟2𝑦𝑦(𝑡)1(𝑡)/𝑦𝑚1𝑦(𝑡)/𝑦𝑛𝑐2𝑦2𝑢(𝑡)1𝑎𝑦(𝑡)𝑧(𝑡)𝑥(𝑡)+𝑎𝑦(𝑡)+𝑏2,𝑡𝑛𝑇,𝑑𝑧(𝑡)𝑢𝑑𝑡=𝑚𝑧(𝑡)+2𝑎𝑦(𝑡)𝑧(𝑡)𝑥(𝑡)+𝑎𝑦(𝑡)+𝑏2+𝑢3𝑥(𝑡)𝑧(𝑡)𝑏1,+𝑥(𝑡)Δ𝑥(𝑡)=0,Δ𝑦(𝑡)=0,𝑡=𝑛𝑇,Δ𝑧(𝑡)=𝛿𝑧(𝑡)+𝑝,(1.1) where 𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡) are the densities of the cyanobacteria population, green algae population, and fish population at time 𝑡, Δ𝑥(𝑡)=𝑥(𝑡+)𝑥(𝑡);Δ𝑦(𝑡)=𝑦(𝑡+)𝑦(𝑡), and Δ𝑧(𝑡)=𝑧(𝑡+)𝑧(𝑡); 𝑟1,𝑟2 is a growth parameter which is related to the biological characteristics of populations and the rationalization of environmental resources; 𝑟1𝐺0  (0𝐺0𝐺1) is the carrying capacity of the cyanobacteria population 𝑥; 𝐺1 is the limiting value of available resources; 𝑦𝑚(0(𝑦𝑚/𝑦𝑛)1) is the maximum density of green algae population 𝑦 (i.e., the environmental carrying capacity); 𝑦𝑛 is a nutritional parameter which is related to the resource conditions of the environment; 𝑢1 is the cropping rate of fish; 𝑢2 and 𝑢3 are the average growth rate of fish that depends on the cyanobacteria and green algae for food; 𝑎 is the capture rate (as a proportion); 𝑚 is the average mortality rate for fish; 𝑏1 and 𝑏2 are the half-saturation constants; 𝑐1and 𝑐2 are the intraspecific competition rates of the cyanobacteria and green algae; 𝑇 is the period of the impulsive effect; 𝑁 is the set of all nonnegative integers; 𝛿(𝛿[0,1]) is the proportion of fish harvested at fixed times 𝑡=𝑛𝑇; and 𝑝>0 is the number of fish released at times 𝑡=𝑛𝑇.

The rest of this paper is arranged as follows: in the next section, some useful notations and basic properties of a system with impulsive effect are given, and then the main theorems and their proofs are developed using the Floquet theory. In Section 3, numerical results will be presented which exhibit the rich dynamic behaviors of system (1.1), and these results will be briefly discussed. Conclusions are given in the last section.

2. Preliminaries and Mathematical Analysis

Let 𝑅+=[0,),𝑅+3={𝑋𝑅3𝑋>0}. Denote by 𝑓=(𝑓1,𝑓2,𝑓3) the mapping defined by the right-hand sides of the first, second, and third equations of system (1.1). Let 𝑉𝑅+×𝑅3+𝑅+; then 𝑉is said to belong to class 𝑉0 if(1)𝑉 is continuous in (𝑛𝑇,(𝑛+1)𝑇]×𝑅3+, and for each 𝑋𝑅3+,𝑛𝑁, lim(𝑡,𝑦)(𝑛𝑇+,𝑋)𝑉(𝑡,𝑦)=𝑉(𝑛𝑇+,𝑋) exists.(2)𝑉 is locally Lipschitzian in 𝑋.

Definition 2.1 (see [18]). Let 𝑉𝑉0; then for (𝑡,𝑥)(𝑛𝑇,(𝑛+1)𝑇]×𝑅3+, the upper right derivative of 𝑉(𝑡) 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+𝑅3+; 𝑋(𝑡) is continuous on (𝑛𝑇,(𝑛+1)𝑇], and 𝑋(𝑛𝑇+)=lim(𝑡𝑛𝑇+)𝑋(𝑛𝑇) exists.
(2) The smoothness properties of 𝑓 guarantee the global existence and uniqueness of solutions of system (1.1) (for details, see [18, 19]).
Because 𝑥(𝑡)=𝑦(𝑡)=𝑧(𝑡)=0 whenever 𝑥(𝑡)=𝑦(𝑡)=𝑧(𝑡)=0   (𝑡𝑛𝑇) and 𝑧(𝑛𝑇+)=(1𝛿)𝑧(𝑛𝑇)+𝑝, the following lemma can be stated.

Lemma 2.3. Assume 𝑋(𝑡) is a solution of system (1.1) such that(1)if𝑋(0+)0, then 𝑋(𝑡)0  for all 𝑡0,(2)if𝑋(0+)>0, then 𝑋(𝑡)>0  for all 𝑡>0.
Next, an important comparison theorem on impulsive differential equations [18] will be stated.

Lemma 2.4 (comparison theorem). Assume that 𝑉𝑉0 and that 𝐷+𝑉𝑡𝑉(𝑡,𝑋)𝑔(𝑡,𝑉(𝑡,𝑋)),𝑡𝑛𝑇,𝑡,𝑋+𝜑𝑛(𝑉(𝑡,𝑋)),𝑡=𝑛𝑇,(2.2) where 𝑔𝑅+×𝑅+𝑅 is continuous on (𝑛𝑇,(𝑛+1)𝑇]×𝑅3+ and for 𝑢𝑅+, 𝑛𝑁, lim(𝑡,𝑦)(𝑛𝑇+,𝑢)𝑔(𝑡,𝑣)=𝑔(𝑛𝑇+,𝑢) exists, with 𝜑𝑛𝑅+𝑅+ nondecreasing. Let 𝑟(𝑡) be the maximal solution of the scalar impulsive differential equation: 𝑑𝑢(𝑡)𝑢𝑡𝑑𝑡=𝑔(𝑡,𝑢(𝑡)),𝑡𝑛𝑇,+=𝜑𝑛𝑢0(𝑢(𝑡)),𝑡=𝑛𝑇,+=𝑢0,(2.3) existing on [0,). Then 𝑉(0+,𝑋0)𝑢0, which implies that 𝑉(𝑡,𝑋(𝑡))𝑟(𝑡), 𝑡0, where 𝑋(𝑡) is any solution of system (1.1).
If the cyanobacteria and green algae are eradicated, then system (1.1) will reduce to the following system: 𝑑𝑧(𝑡)𝑧𝑡𝑑𝑡=𝑚𝑧(𝑡),𝑡𝑛𝑇,+𝑧0=(1𝛿)𝑧(𝑡)+𝑝,𝑡=𝑛𝑇,+=𝑧0.(2.4)

Therefore, the following result can be easily obtained.

Lemma 2.5. 𝑧(𝑡)=𝑝exp(𝑚(𝑡𝑛𝑇))/(1(1𝛿)exp(𝑚𝑇)) is a positive periodic solution of (2.4) with initial value 𝑧(0+)=𝑝/(1(1𝛿)exp(𝑚𝑇)),𝑡(𝑛𝑇,(𝑛+1)𝑇],𝑛𝑁.
𝑧(𝑡)=(𝑧(0+)𝑝/(1(1𝛿)exp(𝑚𝑇)))exp(𝑚𝑡)+𝑧(𝑡) is a general solution of (2.4) with initial value 𝑧0=𝑧(0+)0, where 𝑡(𝑛𝑇,(𝑛+1)𝑇], 𝑛𝑁.
For a positive periodic solution 𝑧(𝑡) of (2.4) and every solution 𝑧(𝑡) of (2.4) with 𝑧00, |𝑧(𝑡)𝑧(𝑡)|0, 𝑡.

From Lemma 2.5, the general solution 𝑧(𝑡) converges to the positive periodic solution 𝑧(𝑡) of (2.4) when 𝑡, and the complete expression for the cyanobacteria and green algae-eradication periodic solution (0,0,𝑧(𝑡)) of system (1.1) becomes0,0,𝑧=(𝑡)0,0,𝑝exp(𝑚(𝑡𝑛𝑇))1(1𝛿)exp(𝑚𝑇),(2.5) for 𝑡(𝑛𝑇,(𝑛+1)𝑇].

Now, the Floquet theorem for impulsive equations [20] will be introduced:𝑑𝑥(𝑡)𝑑𝑡=𝐴(𝑡)𝑥(𝑡),𝑡𝑡𝑘,𝑥𝑡+=𝑥(𝑡)+𝐵𝑘𝑥(𝑡),𝑡=𝑡𝑘,(2.6) with conditions as follows(1)𝐴()PC(𝑅,𝐶𝑛×𝑛) and 𝐴(𝑡+𝑇)=𝐴(𝑡)(𝑡𝑅), where PC(𝑅,𝐶𝑛×𝑛) is the set of all piecewise continuous matrix functions and 𝐶𝑛×𝑛 is the set of 𝑛×𝑛 matrices.(2)𝐵𝑘(𝐶𝑛×𝑛) and det(𝐸+𝐵𝑘)0, 𝑡𝑘𝑇𝑘+1.(3)There exists a 𝑞𝑁 such that 𝐵𝑘+𝑞=𝐵𝑘, 𝑡𝑘+𝑞=𝑡𝑘+𝑇.

Let 𝐻(𝑡) be a fundamental matrix of system (2.6), then there exists a unique nonsingular matrix 𝑀𝐶𝑛×𝑛 such that 𝐻(𝑡+𝑇)=𝐻(𝑡)𝑀. 𝑀 is similar to the matrix of system (2.6) and has the same eigenvalues 𝑢1,𝑢2,,𝑢𝑛. These eigenvalues 𝑢1,𝑢2,,𝑢𝑛 of system (2.6) are called Floquet multipliers.

Lemma 2.6 (Floquet theorem [20]). Assume that (1)–(3) hold, then system (2.6) is(1)stable if and only if 𝑢𝑖1(𝑖=1,,𝑛), and to those 𝑢𝑖=1, there correspond simple elementary divisors;(2)asymptotically stable if and only if all 𝑢𝑖<1(𝑖=1,,𝑛);(3)unstable if 𝑢𝑖>1 for some 𝑖=1,,𝑛.

After these preliminaries, the main theorems of this paper will now be presented, then it will be proven that each solution of system (1.1) is ultimately bounded.

Theorem 2.7. There exists a positive constant 𝑀 such that 𝑥(𝑡)𝑀, 𝑦(𝑡)𝑀, and 𝑧(𝑡)𝑀 for each solution of system (1.1) with all 𝑡 large enough.

Proof. Define 𝑉(𝑡,𝑥(𝑡)) such that 𝑉(𝑡,𝑥(𝑡))=𝑥(𝑡)+𝑢2𝑦(𝑡)+𝑢1𝑧(𝑡).(2.7)
Because (𝑑𝑥(𝑡)/𝑑𝑡)𝑟1𝑥(𝑡)𝑐1𝑥2(𝑡), then 𝑥(𝑡)(𝑟1/𝑐1), and the upper right derivative of 𝑉(𝑡,𝑥(𝑡)) can be calculated along a solution of system (1.1) when 𝑡𝑛𝑇: 𝐷+(𝑉(𝑡))+𝐿𝑉(𝑡)𝐿+𝑟1𝑥(𝑡)𝑐1𝑥2(𝑡)+𝐿𝑢2+𝑢2𝑟2𝑦(𝑡)𝑐2𝑢2𝑦2(𝑡)+𝐿𝑢1𝑢1𝑢𝑚+1𝑟1𝑢3𝑏1𝑐1𝑧(𝑡).(2.8)
Let0<𝐿<𝑚(𝑟1𝑢3/𝑏1𝑐1), then (𝐿+𝑟1)𝑥(𝑡)𝑐1𝑥2(𝑡), (𝐿𝑢2+𝑢2𝑟2)𝑦(𝑡)𝑐2𝑢2𝑦2(𝑡) are both bounded, so 𝐷+(𝑉(𝑡))+𝐿𝑉(𝑡) is bounded, and 𝐿1 and 𝐿2 can be chosen such that 𝑑𝐷+(𝑉(𝑡))𝑑𝑡𝐿1𝑉(𝑡)+𝐿2𝑉𝑡,𝑡𝑛𝑇,+𝑉(𝑡)+𝑢1𝑝,𝑡=𝑛𝑇,(2.9) where 𝐿1 and 𝐿2 are positive constants. According to Lemma 2.4, 𝑉0𝑉(𝑡)+𝐿2𝐿1exp𝐿1𝑡+𝑢1𝑝1exp𝐿1𝑛𝑇1exp𝐿1𝑇exp𝐿1+𝐿(𝑡𝑛𝑇)2𝐿1,(2.10) and hence 𝐿𝑉(𝑡)2𝐿1+𝑢1𝐿𝑝exp1𝑇𝐿exp1𝑇1,(2.11) as 𝑡, so 𝑉(𝑡,𝑥(𝑡)) is ultimately bounded. Therefore, each positive solution of system (1.1) is ultimately bounded, and the proof is completed.

Theorem 2.8. If 𝑟1𝐺0𝐺1𝑢𝑇<1𝑝(1exp(𝑚𝑇))𝑚𝑏1,𝑟(1(1𝛿)exp(𝑚𝑇))2𝑢𝑇<1𝑎𝑝(1exp(𝑚𝑇))𝑚𝑏2,(1(1𝛿)exp(𝑚𝑇))(2.12) then the periodic solution (0,0,𝑧(𝑡)) is locally asymptotically stable.

Proof. The local stability of the periodic solution (0,0,𝑧(𝑡)) may be determined by considering the behavior of small-amplitude perturbations of the solution.
Define 𝑥(𝑡)=𝑢(𝑡),𝑦(𝑡)=𝑣(𝑡),𝑧(𝑡)=𝑤(𝑡)+𝑧(𝑡).
Then the linearization of system (1.1) becomes 𝑑𝑢(𝑡)=𝑟𝑑𝑡1𝐺0𝐺1𝑢1𝑧(𝑡)𝑏1𝑢(𝑡),𝑑𝑣(𝑡)=𝑟𝑑𝑡2𝑢1𝑎𝑧(𝑡)𝑏2𝑣(𝑡),𝑡𝑛𝑇,𝑑𝑤(𝑡)=𝑢𝑑𝑡3𝑧(𝑡)𝑏1𝑢𝑢(𝑡)+2𝑎𝑧(𝑡)𝑏2𝑣(𝑡)𝑚𝑤(𝑡),Δ𝑢(𝑡)=0,Δ𝑣(𝑡)=𝛿𝑣(𝑡),𝑡=𝑛𝑇,Δ𝑤(𝑡)=0,(2.13) and, as a result, 𝑢(𝑡)𝑣(𝑡)𝑤(𝑡)=Φ(𝑡)𝑢(0)𝑣(0)𝑤(0),0𝑡𝑇,(2.14) where Φ(𝑡) satisfies 𝑑Φ(𝑡)=𝑟𝑑𝑡1𝐺0𝐺1𝑢1𝑧(𝑡)𝑏1000𝑟2𝑢1𝑎𝑧(𝑡)𝑏20𝑢3𝑧(𝑡)𝑏1𝑢2𝑎𝑧(𝑡)𝑏2𝑚Φ(𝑡),(2.15)Φ(0)=𝐼, the identity matrix, and 𝑢𝑛𝑇+𝑣𝑛𝑇+𝑤𝑛𝑇+=100010001𝛿𝑢(𝑛𝑇)𝑣(𝑛𝑇)𝑤(𝑛𝑇).(2.16)
Therefore, the stability of the periodic solution (0,0,𝑧(𝑡)) is determined by the eigenvalues 𝜃=100010001𝛿Φ(𝑇).(2.17)
Therefore, all eigenvalues of 𝜃 are given by 𝜆1=exp𝑇0𝑟1𝐺0𝐺1𝑢1𝑧(𝑡)𝑏1,𝜆𝑑𝑡2=exp𝑇0𝑟2𝑢1𝑎𝑧(𝑡)𝑏2,𝜆𝑑𝑡3=(1𝛿)exp(𝑚𝑇)<1.(2.18)
According to Lemma 2.6, (0,0,𝑧(𝑡)) is locally asymptotically stable if 𝜆1<1,𝜆2<1, that is to say 𝑟1𝐺0𝐺1𝑢𝑇<1𝑝(1exp(𝑚𝑇))𝑚𝑏1,𝑟(1(1𝛿)exp(𝑚𝑇))2𝑢𝑇<1𝑎𝑝(1exp(𝑚𝑇))𝑚𝑏2,(1(1𝛿)exp(𝑚𝑇))(2.19) which completes the proof.

Theorem 2.9. If 𝑟1𝐺0𝐺1𝑢𝑇<1𝑝(1exp(𝑚𝑇))𝑚𝑏1,𝑟(1(1𝛿)exp(𝑚𝑇))2𝑢𝑇<1𝑎𝑝(1exp(𝑚𝑇))𝑚𝑏2,𝑟(1(1𝛿)exp(𝑚𝑇))1𝑢1𝑏1+𝑀𝑝exp(𝑚𝑇)𝑟1(1𝛿)exp(𝑚𝑇)<0,2𝑢1𝑎𝑀+𝑎𝑀+𝑏2𝑝exp(𝑚𝑇)1(1𝛿)exp(𝑚𝑇)<0,(2.20) then the periodic solution (0,0,𝑧(𝑡)) is globally asymptotically stable.

Proof. From Theorem 2.8, the periodic solution (0,0,𝑧(𝑡)) is locally asymptotically stable. Now it will be proved to be a global attractor. Let 𝑉(𝑡)=𝑥(𝑡)+𝑦(𝑡), then by Theorem 2.7, there exists a constant 𝑀>0 such that 𝑥(𝑡)𝑀, 𝑦(𝑡)𝑀, and 𝑧(𝑡)𝑀. Obviously, 𝑉||(1.1)𝑟𝑥(𝑡)1𝑢1𝑧(𝑡)𝑏1+𝑀𝑐1𝑥2𝑟(𝑡)+𝑦(𝑡)2𝑢1𝑎𝑧(𝑡)𝑀+𝑎𝑀+𝑏2𝑐2𝑦2(𝑡).(2.21)
However, for system (1.1), 𝑑𝑧(𝑡)𝑧𝑡𝑑𝑡𝑚𝑧(𝑡),𝑡𝑛𝑇,+=(1𝛿)𝑧(𝑡)+𝑝,𝑡=𝑛𝑇.(2.22)
Therefore, there exists a 𝜏>0, and an 𝜀>0 small enough, such that 𝑧(𝑡)𝑧1(𝑡)𝜀 for all 𝑡>𝜏. This leads to 𝑧(𝑡)𝑧1(𝑡)𝜀=𝑝exp(𝑚𝑇)1(1𝛿)exp(𝑚𝑇)𝜀.(2.23)
Let 𝛾Δ=𝑝exp(𝑚𝑇)/(1(1𝛿)exp(𝑚𝑇))𝜀; it is well known that 𝑟1(𝑢1𝛾/(𝑏1+𝑀))<0 and 𝑟2(𝑢1𝑎𝛾/(𝑀+𝑎𝑀+𝑏2))<0. Therefore, when 𝑡>𝜏, 𝑉||(1.1)𝑟𝑥(𝑡)1𝑢1𝑧(𝑡)𝑏1+𝑀𝑐1𝑥2𝑟(𝑡)+𝑦(𝑡)2𝑢1𝑎𝑧(𝑡)𝑀+𝑎𝑀+𝑏2𝑐2𝑦2(𝑡)<0.(2.24)
Therefore, 𝑉(𝑡)0, 𝑥(𝑡)0, 𝑦(𝑡)0 as 𝑡, and it follows that the periodic solution (0,0,𝑧(𝑡)) is a global attractor. This completes the proof.

Theorem 2.10. The system (1.1) is permanent, if 𝑟1𝐺0𝐺1𝑢𝑇>1𝑝(1exp(𝑚𝑇))𝑚𝑏1,𝑟(1(1𝛿)exp(𝑚𝑇))2𝑢𝑇>1𝑎𝑝(1exp(𝑚𝑇))𝑚𝑏2,𝑟(1(1𝛿)exp(𝑚𝑇))1𝑢1𝑏1+𝑀𝑝exp(𝑚𝑇)𝑟1(1𝛿)exp(𝑚𝑇)>0,2𝑢1𝑎𝑀+𝑎𝑀+𝑏2𝑝exp(𝑚𝑇)1(1𝛿)exp(𝑚𝑇)>0.(2.25)

Proof. Assume that 𝑋(𝑡)=(𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)) is any solution of system (1.1) with 𝑋(0)>0. From Theorem 2.7, 𝑥(𝑡)𝑀, 𝑦(𝑡)𝑀, and 𝑧(𝑡)𝑀 with 𝑡0. From Lemma 2.5, 𝑧(𝑡)𝑧(𝑡)𝜀 for all 𝑡 large enough and some 𝑡 with 𝑧(𝑡)>𝑝exp(𝑚𝑇)/(1(1𝛿)exp(𝑚𝑇))𝜀=𝜉1. Therefore, it is necessary to find 𝜉2 and 𝜉3 such that 𝑥(𝑡)𝜉2 and 𝑦(𝑡)𝜉3 for 𝑡 large enough. Select an 𝜀1 small enough so that 𝜑1=exp(𝑛+1)𝑇𝑛𝑇𝑟1𝐺0𝐺1𝑟1𝑀𝐺1𝑀𝑐1𝑢𝑀1𝑏1𝑣(𝑡)+𝜀1𝑑𝑡>1.(2.26)
Next, it is necessary to prove that 𝑥(𝑡)𝜉2 for 𝑡 large enough, where 𝜉2 is a positive constant.
It is easy to prove that there exists a 𝑡1(0,) such that 𝑥(𝑡)𝜉2. Otherwise, if 𝑥(𝑡)<𝜉2 for all 𝑡>0, 𝑑𝑧(𝑡)𝑢𝑑𝑡2𝑎𝑀𝑏2+𝑢3𝑀𝑏1𝑧𝑡𝑚𝑧(𝑡),𝑡𝑛𝑇,+𝑧0=(1𝛿)𝑧(𝑡)+𝑝,𝑡=𝑛𝑇,+=𝑧0,(2.27) and therefore 𝑧(𝑡)𝑣(𝑡) and 𝑣(𝑡)𝑣(𝑡), where 𝑣(𝑡) is the solution of 𝑑𝑣(𝑡)=𝑢𝑑𝑡2𝑎𝑀𝑏2+𝑢3𝑀𝑏1𝑣𝑡𝑚𝑣(𝑡),𝑡𝑛𝑇,+𝑣0=(1𝛿)𝑣(𝑡)+𝑝,𝑡=𝑛𝑇,+=𝑣0,(2.28) with 𝑣𝑢(𝑡)=𝑝exp𝑚2𝑎𝑀/𝑏2𝑢3𝑀/𝑏1(𝑡𝑛𝑇)𝑢1(1𝛿)exp𝑚2𝑎𝑀/𝑏2𝑢3𝑀/𝑏1𝑇],𝑡(𝑛𝑇,(𝑛+1)𝑇.(2.29)
Therefore, there exists a 𝑇1>0 such that 𝑧(𝑡)𝑣(𝑡)𝑧(𝑡)+𝜀1, and 𝑑𝑥(𝑡)𝑟𝑑𝑡𝑥(𝑡)1𝐺0𝐺1𝑟1𝑀𝐺1𝑀𝑐1𝑢𝑀1𝑏1𝑣(𝑡)+𝜀1,(2.30) for 𝑡>𝑇1. Let 𝑁1𝑁 and 𝑁1𝑇𝑇2𝑇1. Integrating (2.30) over (𝑛𝑇,(𝑛+1)𝑇]𝑛>𝑁1 yields 𝑥((𝑛+1)𝑇)𝑥𝑛𝑇+exp(𝑛+1)𝑇𝑛𝑇𝑟1𝐺0𝐺1𝑟1𝑀𝐺1𝑀𝑐1𝑢𝑀1𝑏1𝑣(𝑡)+𝜀1𝑑𝑡=𝑥𝑛𝑇+,exp(𝑛+1)𝑇𝑛𝑇𝑟1𝐺0𝐺1𝑟1𝑀𝐺1𝑀𝑐1𝑢𝑀1𝑏1𝑣(𝑡)+𝜀1𝑑𝑡=𝑥𝑛𝑇+𝜑1.(2.31)
Therefore, 𝑥((𝑁1+𝑛)𝑇)𝑥(𝑁1𝑇)𝜑𝑛1 as 𝑛, but this leads to a contradiction because 𝑥(𝑡) is bounded. Therefore, there exists a 𝑡1>0 such that 𝑥(𝑡1)𝜉2.
Second, if 𝑥(𝑡1)𝜉2 for all 𝑡𝑡1, then the proof is complete. Hence, it is only necessary to consider those solutions which leave the region 𝑅={𝑥(𝑡)𝑥(𝑡)<𝜉2} and reenter it. Let 𝑡(𝑡)=inf𝑡𝑡1{𝑥(𝑡)<𝜉2}, then 𝑥(𝑡)𝜉2, 𝑡(𝑡,𝑡), and 𝑡(𝑛1𝑇,(𝑛1+1)𝑇). However, obviously 𝑥(𝑡)=𝜉2 because 𝑋(𝑡)=(𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)) is continuous. It is claimed here that there must exist a 𝑡2((𝑛1+1)𝑇,(𝑛1+1)𝑇+𝑇1) such that 𝑥(𝑡2)𝜉2. Otherwise, 𝑥(𝑡)<𝜉2, for 𝑡((𝑛1+1)𝑇,(𝑛1+1)𝑇+𝑇1),𝑇1=𝑛2𝑇+𝑛3𝑇, 𝑛2,𝑛3𝑁, and then 𝑛2𝜀1𝑇>ln1/(𝑀+𝑝)𝑢2𝑎𝑀/𝑏2+𝑢3𝑀/𝑏1,𝜑𝑚exp2𝑛2𝑇𝜑+1𝑛31>1.(2.32)
Consider (2.28) with 𝑣(𝑡+)=𝑧(𝑡+); this leads to 𝑣𝑛𝑣(𝑡)=1𝑇+1+𝑝𝑢1(1𝛿)exp2𝑎𝑀/𝑏2+𝑢3𝑀/𝑏1𝑇𝑢𝑚×exp2𝑎𝑀𝑏2+𝑢3𝑀𝑏1𝑛𝑚𝑡1𝑇+1+𝑣(𝑡),(2.33) for 𝑡(𝑛𝑇,(𝑛+1)𝑇], 𝑛1+1<𝑛<𝑛1+𝑛2+𝑛3+1, and then ||𝑣(𝑡)𝑣||𝑢(𝑡)<(𝑀+𝑝)exp2𝑎𝑀𝑏2+𝑢3𝑀𝑏1𝑛𝑚𝑡1𝑇+1<𝜀1,𝑧(𝑡)𝑣(𝑡)𝑣(𝑡)+𝜀1,(2.34) for 𝑡[(𝑛1+𝑛2+1)𝑇,(𝑛1+1)𝑇+𝑇1], which implies (2.30). Integrating (2.30) over 𝑡[(𝑛1+𝑛2+1)𝑇,(𝑛1+1)𝑇+𝑇1] yields 𝑥𝑛1+𝑛2+𝑛3𝑇𝑛+1𝑥1+𝑛2𝑇𝜑+1𝑛31.(2.35)
For every 𝑡(𝑡,(𝑛1+1)𝑇), there are only two possible cases: one, if 𝑥(𝑡)<𝜉2 for 𝑡(𝑡,(𝑛1+1)𝑇), when 𝑡(𝑡,(𝑛1+𝑛2+1)𝑇), then 𝑥(𝑡)<𝜉2, and hence 𝑑𝑥(𝑡)𝑟𝑑𝑡𝑥(𝑡)1𝐺0𝐺1𝑟1𝜉2𝐺1𝑀𝑐1𝑢𝑀1𝑏1𝑀=𝜑2𝑥(𝑡).(2.36)
Integrating (2.36) over 𝑡(𝑡,(𝑛1+𝑛2+1)𝑇),𝑥((𝑛1+𝑛2+1)𝑇)𝑥(𝑡)exp(𝜑2(𝑛2+1)𝑇) is obvious.
Therefore,𝑥((𝑛1+𝑛2+𝑛3+1)𝑇)𝜉2exp(𝜑2(𝑛2+1)𝑇)𝜑𝑛31>𝜉2, but this leads to a contradiction. Hence, let 𝑡3=inf𝑡>𝑡{𝑥(𝑡)𝜉2}, then 𝑥(𝑡3)=𝜉2, and (2.36) holds. Integrating (2.36) over [𝑡,𝑡3), 𝑥𝑡(𝑡)𝑥𝜑exp2𝑡𝑡𝜉2𝜑exp2𝑛1+𝑛3𝑇+1=𝜉5.(2.37)
𝑥(𝑡3)𝜉2 is also true for 𝑡>𝑡3. Hence, 𝑥(𝑡)𝜉5, 𝑡>𝑡3. Two, there exists a 𝑡5(𝑡,(𝑛1+1)𝑇] such that 𝑥(𝑡5)𝜉2. Therefore, let 𝑡4=inf𝑡>𝑡{𝑥(𝑡)𝜉2}, then 𝑥(𝑡)<𝜉2 for 𝑡[𝑡,𝑡4) and 𝑥(𝑡4)=𝜉2. For 𝑡[𝑡,𝑡4), (2.36) holds, and integrating it over 𝑡[𝑡,𝑡4), 𝑥(𝑡)𝑥(𝑡)exp(𝜑2(𝑡𝑡))>𝜉5. This process can be continued because 𝑥(𝑡4)𝜉2 and 𝑥(𝑡)𝜉5, 𝑡>𝑡4. This yields the two possible cases, 𝑥(𝑡)𝜉5, 𝑡𝑡1, and 𝑦(𝑡)𝜉4, 𝑡𝑡2. Let Ω={(𝑥,𝑦,𝑧)𝑥(𝑡)𝜉2,𝑦(𝑡)𝜉3,𝑧(𝑡)𝜉1,𝑥(𝑡)+𝑦(𝑡)+𝑧(𝑡)3𝑀}. Then the set Ω is a global attractor. Moreover, every solution 𝑋(𝑡)=(𝑥(𝑡),𝑦(𝑡),𝑧(𝑡)) of system (1.1) will eventually enter and remain in the set Ω. Therefore, system (1.1) is permanent, and the proof is completed.

3. Numerical Analysis

3.1. Bifurcation Analysis

In Section 2, it has been proved that the cyanobacteria and green algae-eradication periodic solution is locally asymptotically stable, and conditions have been established for the permanence of system (1.1). Now, to substantiate these theoretical results, the following parameters will be considered: 𝑢1=0.5, 𝑢2=0.6, 𝑢3= 0.4, 𝑟1=0.5, 𝑟2=0.5, 𝐺0=10, 𝐺1=15, 𝑦𝑚=15, 𝑦𝑛=20, 𝑐1=0.1, 𝑐2=0.12, 𝑎=0.75, 𝑏1=0.45, 𝑏2=0.6, 𝛿=0.55, 𝑇=40 with initial values 𝑥0=0.5, 𝑦0=0.6, 𝑧0=0.7.

From Theorem 2.8, it is known that the periodic solution (0,0,𝑧(𝑡)) is locally asymptotically stable if 𝑝>𝑝max=8.15; this periodic solution (0,0,𝑧(𝑡)) is shown in Figure 1. It is clear that the variable fish 𝑧 oscillates in a stable cycle, but the cyanobacteria population 𝑥 and green algae population 𝑦 rapidly decrease to zero when 𝑝>𝑝max. When 𝑝 is smaller than 𝑝max, then the cyanobacteria and green algae-eradication periodic solution will become unstable, and it is possible that the cyanobacteria population, green algae population, and fish population can coexist.

fig1
Figure 1: Dynamic behavior of system (1.1). When 𝑝>𝑝max=8.15, the cyanobacteria and green algae will become extinct. Time series evolving according to biological control system (1.1): (a) the cyanobacteria population 𝑥, (b) the green algae population 𝑦, and (c) the fish population 𝑧.

Figure 2 shows typical bifurcation diagrams for system (1.1) with respect to 𝑝 in the range 𝑝[0,10]. As 𝑝 increases, system (1.1) exhibits rich dynamic behavior, such as chaotic bands with periodic windows, crises, period-halving bifurcations, quasi-periodic oscillations, and narrow or wide periodic windows. When 𝑝<0.89, the cyanobacteria population 𝑥 enters a chaotic band. As 𝑝 increases beyond 0.89, the cyanobacteria population 𝑥 enters a chaotic band with periodic windows, crisis phenomena appear, and the chaotic attractor suddenly changes into a periodic attractor. Then the periodic attractor loses its stability, and system (1.1) again enters a chaotic band. Finally, the chaotic attractor changes into a periodic attractor again; details of these results are shown in Figure 3. When 1.15<𝑝<2.7, there is a cascade of period-halving bifurcation which leads system (1.1) to a 𝑇-periodic solution (Figure 4), where the period-halving bifurcation is the opposite of the bifurcation observed earlier. When 𝑝>2.75, the green algae population 𝑦 decreases to zero because of the increasing number of fish population 𝑧 and the competition between the cyanobacteria population 𝑥 and green algae population 𝑦. Then the cyanobacteria population 𝑥 and fish population 𝑧 can coexist in a stable cycle, but the numbers of the cyanobacteria population 𝑥 will decrease because of short supply of resources. When 𝑝>𝑝max= 8.15 the cyanobacteria population 𝑥 will be eradicated because the number of fish population 𝑧 released is continually increasing, then the cyanobacteria and green algae-eradication periodic solution occurs. Once the cyanobacteria population𝑥 and green algae population 𝑦 have decreased to zero, fish population 𝑧 will be eradicated after a period of time because of lack of food. Therefore, it is apparent that the value of 𝑝 can effectively control the size of the cyanobacteria population and green algae population. All these numerical simulations are consistent with the theoretical proofs presented earlier.

fig2
Figure 2: Bifurcation diagrams for system (1.1) showing the effect of 𝑝. (a) 𝑥 versus 𝑝, (b) 𝑦 versus 𝑝, and (c) 𝑧 versus 𝑝.
fig3
Figure 3: Crisis. (a) Chaotic attractor when 𝑝=0.88, (b) phase diagram of periodic solution when𝑝=0.9, (c) chaotic attractor when 𝑝=1, and (d) phase diagram of periodic solution when𝑝=1.1.
fig4
Figure 4: Period-halving bifurcation. (a) Chaotic attractor when 𝑝=1.3, (b) phase diagram of a 4T-periodic solution when 𝑝=1.55, (c) phase diagram of a 2T-periodic solution when𝑝=2, and (d) a period-T attractor when𝑝=2.5.
3.2. Strange Attractors and Power Spectra

Now the commonly used method called power spectra will be used to study the qualitative nature of strange attractors [21]. By calculating the largest Lyapunov exponent for strange attractor (a) (Figure 5(a)), this value is determined to be 0.18219. Obviously, the strange attractor is a chaotic attractor. Furthermore, the spectrum of chaotic attractor (a) (Figure 5(b)) is composed of dense wave bands and sparse wave peaks with low wave troughs. These results agree with the observation that the chaotic attractor comes into being because some cycles lose weak stability.

fig5
Figure 5: Strange attractor and power spectrum. (a) Strange attractor when 𝑝=1.25, (b) power spectrum of attractor (a).
3.3. The Largest Lyapunov Exponent

Convincing evidence for deterministic chaos has come from several recent experiments [2224]. From these results, it is clear that chaos plays a very important role in these studies, and therefore detecting and exploring chaos is very important [2531]. Therefore, the largest Lyapunov exponent is considered to be the most useful diagnostic tool for chaotic systems. 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. The largest Lyapunov exponent for system (1.1) was calculated for various values of 𝑝, and Figure 6 shows the results for 𝑝 from 0 to 2.

452789.fig.006
Figure 6: The largest Lyapunov exponents (LLE) for system (1.1) with 𝑝 varying between 0 and 2.

4. Conclusions

In this paper, the effects of impulsive perturbations on a fish algae model have been investigated. It has been proved by means of the Floquet theory of impulsive differential equations and small-amplitude perturbation techniques that system (1.1) has a stable algae-eradication periodic solution. Furthermore, the conditions for system (1.1) to be permanent have been determined using the comparison theorem. Typical bifurcation diagrams have been analyzed in detail, revealing that the system exhibits very rich dynamics. From Theorem 2.8, it is clear that the cyanobacteria and green algae-eradication periodic solution (0,0,𝑧(𝑡)) is locally asymptotically stable if 𝑝>𝑝max=8.15, but that the cyanobacteria population 𝑥 and green algae population 𝑦 rapidly decrease to zero. When the value of 𝑝>2.75, green algae population 𝑦 decreases to zero, but the cyanobacteria population 𝑥 and fish population 𝑧 can coexist in a cycle; when 2.25<𝑝<2.75, the cyanobacteria population, green algae population and fish population can coexist in a cycle. When 0<𝑝<2.25, system (1.1) undergoes complex dynamics. From the above analysis, it is apparent that the numbers of both algae species can be controlled effectively using an impulsive control strategy.

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. 31170338 and Grant no. 30970305).

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