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

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 [8–11], which make the models more reasonable [12–14]. 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+𝐿−1)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=𝑢^𝑐𝑥(𝑡)1−𝑥(𝑡)/𝑥^𝑚1−𝑥(𝑡)/𝑥_ℎ−𝑐¹𝑥²(𝑡)−𝑢¹𝑥(𝑡)𝑦(𝑡)𝑑𝑦(𝑡)𝑥(𝑡)+𝑘₁,𝑑𝑡=−𝑢³𝑦(𝑡)+𝑢²𝑥(𝑡)𝑦(𝑡)𝑥(𝑡)+Δ𝑥𝑘₁(𝑡),=−𝛿₁𝑡≠𝑛𝑇,𝑡≠(𝑛+𝐿−1)𝑇,Δ𝑦(𝑡)=−𝛿₂𝑥(𝑡),ΔΔ𝑥𝑦((𝑦𝑡𝑡))(𝑡==),0𝑝,,𝑡=𝑡=(𝑛𝑛+𝑇,𝐿−1)𝑇,(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=𝑢_𝑐𝑥(𝑡)((1−(𝑥(𝑡)/𝑥_𝑚))/(1−(𝑥(𝑡)/𝑥_ℎ))) is a mathematical model for a single population [17] and is established by Cui and Lawson; 𝑢_𝑐 is a growth parameter which is related to the biological characteristics of populations and the rationalization of environmental resources; 𝑥_𝑚(0≤𝑥_𝑚/𝑥_ℎ≤1) is the maximum density of the algae population (i.e., environmental carrying capacity); 𝑥_ℎ 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; 0≤𝛿₁, 𝛿₂≤1 represent the fraction of the algae and fish which die due to the harvesting or chemical poisoning at𝑡=(𝑛+𝐿−1)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⁺={𝑋∈𝑅²∣𝑋>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 𝑉∶𝑅₊×𝑅²⁺→𝑅₊, then 𝑉 is said to belong to class 𝑉₀ if(1)𝑉 is continuous in ((𝑛−1)𝑇,(𝑛+𝑙−1)𝑇]×𝑅²⁺, ((𝑛+𝑙−1)𝑇,𝑛𝑇]×𝑅²⁺, and for each 𝑋∈𝑅²⁺,𝑛∈𝑁,lim_{(𝑡,𝑦)→((𝑛+𝑙−1)𝑇}⁺_,𝑋)𝑉(𝑡,𝑦)=𝑉((𝑛+𝑙−1)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 𝑉∈𝑉₀; for (𝑡,𝑥)∈((𝑛−1)𝑇,(𝑛+𝑙−1)𝑇]×𝑅²⁺ and ((𝑛+𝑙−1)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)=lim_ℎ→0+sup1ℎ[𝑉(𝑡+ℎ,𝑋+ℎ𝑓(𝑡,𝑋))−𝑉(𝑡,𝑋)].(2.1)

Remark 2.2. (1) The solution of system (1.1) is a piecewise continuous function with 𝑋∶𝑅₊→𝑅²⁺, then 𝑋(𝑡) is continuous on ((𝑛−1)𝑇,(𝑛+𝑙−1)𝑇), and ((𝑛+𝑙−1)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 𝑡≥0,(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=−𝑢³𝑦𝑡𝑦(𝑡),𝑡≠(𝑛+𝑙−1)𝑇,𝑡≠𝑛𝑇,₊=𝑦1𝑡−𝛿₂𝑦(𝑡),𝑡=(𝑛+𝑙−1)𝑇,₊=𝑦𝑦0(𝑡)+𝑝,𝑡=𝑛𝑇,₊=𝑦₀.(2.2)

System (2.2) is a periodically forced linear system, then we get that𝑦^∗(𝑡)=⎧⎪⎪⎨⎪⎪⎩𝑝exp−𝑢³(𝑡−(𝑛−1)𝑇)𝑝1−1−1𝛿−₂𝛿₂exp−𝑢₃𝑇,(𝑛−1)𝑇<𝑡≤(𝑛+𝑙−1)𝑇,exp−𝑢₃(𝑡−(𝑛−1)𝑇)1−1−𝛿₂exp−𝑢₃𝑇,(𝑛+𝑙−1)𝑇<𝑡≤𝑛𝑇,(2.3) is a positive periodic solution of system (2.2) with the initial values 𝑦^∗0⁺=𝑦^∗𝑛𝑇⁺=𝑝1−1−𝛿₂exp−𝑢₃𝑇,𝑦^∗𝑙𝑇⁺=𝑝1−𝛿²exp−𝑢₃𝑙𝑇1−1−𝛿₂exp−𝑢₃𝑇,(2.4) since the general solution of (2.2) is 𝑦(𝑡)=⎧⎪⎪⎪⎨⎪⎪⎪⎩1−𝛿²^𝑛−1𝑦0⁺−𝑝1−1−𝛿₂exp−𝑢₃𝑇exp−𝑢³1−𝛿₂(𝑛−1)𝑇<𝑡≤(𝑛𝑇+𝑙+−𝑦^∗1)(𝑡𝑇),,^𝑛𝑦0⁺−𝑝1−1−𝛿₂exp−𝑢₃𝑇exp−𝑢³(𝑛+𝑙−1)𝑇𝑇<+𝑡𝑦≤^∗𝑛(𝑡𝑇).,(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 𝑢_𝑐𝑇−𝑢¹𝑝1−𝛿₂exp−𝑢₃𝑙𝑇−1−𝛿₂exp−𝑢₃𝑇𝑢₃𝑘₁1−1−𝛿₂exp−𝑢₃𝑇<ln1−1𝛿₁,(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)𝑑𝑡=𝑢^𝑐−𝑢¹𝑦^∗(𝑡)𝑑𝑣(𝑡)𝑘₁𝑢(𝑡),𝑑𝑡=−𝑢³𝑣(𝑡)+𝑢²𝑢(𝑡)𝑦^∗(𝑡)𝑘₁Δ𝑢(𝑡)=−𝛿₁,𝑡≠𝑛𝑇,𝑡≠(𝑛+𝐿−1)𝑇,Δ𝑣(𝑡)=−𝛿₂𝑢(𝑡),ΔΔ𝑢𝑣((𝑣𝑡𝑡))(𝑡==),00,,𝑡=𝑡=(𝑛𝑛+𝑇,𝐿−1)𝑇,(2.7) and as a result, ⎛⎜⎜⎝𝑣𝑢((𝑡𝑡))⎞⎟⎟⎠=Φ(𝑡)⎛⎜⎜⎝𝑣𝑢((00))⎞⎟⎟⎠,0≤𝑡<𝑇,(2.8) where Φ(𝑡) satisfies 𝑑Φ(𝑡)𝑑𝑡=⎛⎜⎜⎜⎝𝑢^𝑐−𝑢¹𝑦^∗(𝑡)𝑘₁𝑢₂0𝑦^∗(𝑡)𝑘₁−𝑢₃⎞⎟⎟⎟⎠Φ(𝑡),(2.9) and Φ(0)=𝐼, the identity matrix. The linearization of the third and fourth equations of (2.2) becomes ⎛⎜⎜⎝𝑣𝑢((𝑛𝑛++𝑙𝑙−−11))𝑇𝑇⁺₊⎞⎟⎟⎠=⎛⎜⎜⎝1−𝛿¹01−0𝛿₂⎞⎟⎟⎠⎛⎜⎜⎝𝑣𝑢((((𝑛𝑛++𝑙𝑙−−11))𝑇𝑇))⎞⎟⎟⎠.(2.10) The linearization of fifth and sixth equations of (2.2) becomes ⎛⎜⎜⎝𝑣𝑢𝑛𝑛𝑇𝑇⁺₊⎞⎟⎟⎠=⎛⎜⎜⎝1001⎞⎟⎟⎠⎛⎜⎜⎝𝑣𝑢((𝑛𝑛𝑇𝑇))⎞⎟⎟⎠.(2.11) The stability of the periodic solution (0,𝑦^∗(𝑡)) is determined by the eigenvalues of 𝜃=⎛⎜⎜⎝1−𝛿¹01−0𝛿₂⎞⎟⎟⎠⎛⎜⎜⎝1001⎞⎟⎟⎠Φ(𝑡).(2.12) Therefore, all eigenvalues of 𝜃 are given by 𝜆₁=1−𝛿₁exp^𝑇0𝑢^𝑐−𝑢¹𝑦^∗(𝑡)𝑘₁𝑑𝑡,𝜆²=1−𝛿₂exp−𝑢₃𝑇<1.(2.13) According to Floquet theory, (0,𝑦^∗(𝑡)) is locally asymptotically stable if 𝜆₁<1, that is to say, 𝑢_𝑐𝑇−𝑢¹𝑝1−𝛿₂exp−𝑢₃𝑙𝑇−1−𝛿₂exp−𝑢₃𝑇𝑢₃𝑘₁1−1−𝛿₂exp−𝑢₃𝑇<ln1−1𝛿₁.(2.14) In the following, we prove the global attractivity. Choose a 𝜀>0 such that 𝜉₁≈1−𝛿₁exp^𝑇0𝑢^𝑐−𝑢¹𝑘₁𝑦^∗(𝑡)−𝜀𝑑𝑡<1,(2.15) and note that 𝑑𝑦(𝑡)/𝑑𝑡≥−𝑢₃𝑦(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 𝑡≥0. From (1.1) and (2.16), 𝑑𝑥(𝑡)𝑑𝑡≤𝑥(𝑡)𝑢^𝑐−𝑢¹𝑘₁𝑦^∗𝑥𝑡(𝑡)−𝜀,𝑡≠(𝑛+𝑙−1)𝑇,₊=1−𝛿₁𝑥(𝑡),𝑡=(𝑛+𝑙−1)𝑇,(2.17) which leads to 𝑥((𝑛+𝑙)𝑇)≤𝑥(𝑛+𝑙−1)𝑇⁺exp^{(𝑛+𝑙)𝑇(𝑛+𝑙−1)𝑇}𝑢^𝑐−𝑢¹𝑘₁𝑦^∗=𝑥((𝑛+𝑙−1)𝑇)1−𝛿₁(𝑡)−𝜀𝑑𝑡=𝑥((𝑛+𝑙−1)𝑇)𝜉₁exp^{(𝑛+𝑙)𝑇(𝑛+𝑙−1)𝑇}𝑢^𝑐−𝑘𝑢¹₁𝑦^∗(𝑡)−𝜀𝑑𝑡.(2.18) Hence, 𝑥((𝑛+𝑙)𝑇)≤𝑥(𝑙𝑇)𝜉^𝑛1 and 𝑥((𝑛+𝑙)𝑇)→0 as 𝑛→∞. Therefore, 𝑥(𝑡)→0 when 𝑛→∞, because 0<𝑥(𝑡)<𝑥((𝑛+𝑙−1)𝑇)(1−𝛿₁)exp(𝑢_𝑐𝑇) for (𝑛+𝑙−1)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,𝑡≥𝑇^. Without any loss of generality, we assume that 0<𝑥(𝑡)<𝜀 for all 𝑡≥0, then from system (1.1),−𝑢₃𝑦(𝑡)≤𝑑𝑦(𝑡)𝑑𝑡≤−𝑢³+𝑢³𝑘₁𝑢₂𝜀𝑦(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=−𝑢³𝑧𝑧₁(𝑡),𝑡≠(𝑛+𝑙−1)𝑇,𝑡≠𝑛𝑇,₁𝑡⁺𝑧=1−𝛿₂𝑧₁(𝑡),𝑡=(𝑛+𝑙−1)𝑇,₁𝑡⁺𝑧=𝑧₁(𝑡)+𝑝,𝑡=𝑛𝑇,₁𝑑𝑧₂0⁺=𝑦0⁺,(2.20)(𝑡)𝑑𝑡=−𝑢³+𝑢³𝑘₁𝑢₂𝑧𝜀𝑧₂(𝑡),𝑡≠(𝑛+𝑙−1)𝑇,𝑡≠𝑛𝑇,₂𝑡⁺𝑧=1−𝛿₂𝑧₂(𝑡),𝑡=(𝑛+𝑙−1)𝑇,₂𝑡⁺𝑧=𝑧₂(𝑡)+𝑝,𝑡=𝑛𝑇,₂0⁺=𝑦0⁺,(2.21) respectively, 𝑧_2∗(𝑡)=⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑝exp−𝑢³+𝑢₃𝑘₁/𝑢₂𝜀(𝑡−(𝑛−1)𝑇)𝑝1−1−1𝛿−₂𝛿₂ex(p𝑛−1−)𝑢𝑇₃<+𝑡𝑢≤₃𝑘(₁𝑛/𝑢+₂𝑙𝜀−1𝑇)𝑇,,exp−𝑢₃+𝑢₃𝑘₁/𝑢₂𝜀(𝑡−(𝑛−1)𝑇)1−1−𝛿₂exp−𝑢₃+(𝑛𝑢+₃𝑘𝑙₁−/𝑢₂1)𝜀𝑇<𝑇𝑡≤𝑛𝑇.,(2.22)
Therefore, for any 𝜀₁>0, there exists a 𝑇₁>0 such that𝑧^∗1(𝑡)−𝜀₁≤𝑦(𝑡)≤𝑧^∗2(𝑡)+𝜀₁,for𝑡>𝑇₁.(2.23) Let 𝜀→0 such that 𝑦^∗(𝑡)−𝜀₁≤𝑦(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 𝑢_𝑐𝑇−𝑢¹𝑝1−𝛿₂exp−𝑢₃𝑙𝑇−1−𝛿₂exp−𝑢₃𝑇𝑢₃𝑘₁1−1−𝛿₂exp−𝑢₃𝑇>ln1−1𝛿₁(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(1−𝛿₂)exp(−𝑢₃𝑇)/(1−(1−𝛿₂)exp(−𝑢₃𝑇))−𝜀≈𝜁₂ 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 𝜓≈𝑢_𝑐1−𝑥𝜁³_𝑚𝑇−𝑐¹𝜁₃𝑇−𝑢¹𝜀₁𝑘₁−𝑢₁𝑇𝑘₁𝑢¹𝑝1−𝛿₂exp−𝑢₃+𝑢₃𝑘₁/𝑢₂𝜁₃𝑙𝑇−1−𝛿₂exp−𝑢₃+𝑢₃𝑘₁/𝑢₂𝜁₃𝑇>1,𝑢₃−𝑢₃𝑘₁/𝑢₂𝜁₃𝑘₁1−1−𝛿₂exp−𝑢₃+𝑢₃𝑘₁/𝑢₂𝜁₃𝑇(2.26) then it is easy to prove that 𝑥(𝑡)<𝜁₃ cannot hold for all 𝑡. Otherwise, 𝑑𝑦(𝑡)𝑑𝑡≤−𝑢³+𝑢³𝑘₁𝑢₂𝜁₃𝑦𝑡𝑦(𝑡),𝑡≠(𝑛+𝑙−1)𝑇,𝑡≠𝑛𝑇,₊=𝑦1𝑡−𝛿₂𝑦(𝑡),𝑡=(𝑛+𝑙−1)𝑇,₊=𝑦𝑦0(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)𝑑𝑡=−𝑢³+𝑢³𝑘₁𝑢₂𝜁₃𝑧𝑡𝑧(𝑡),𝑡≠(𝑛+𝑙−1)𝑇,𝑡≠𝑛𝑇,₊=𝑧1𝑡−𝛿₂𝑧(𝑡),𝑡=(𝑛+𝑙−1)𝑇,₊𝑧=0𝑧(𝑡)+𝑝,𝑡=𝑛𝑇,₊𝑧=𝑦0⁺,(2.28)_∗(𝑡)=⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑝exp−𝑢³+𝑢₃𝑘₁/𝑢₂𝜀(𝑡−(𝑛−1)𝑇)𝑝1−1−1𝛿−₂𝛿₂ex(p𝑛−1−)𝑢𝑇₃<+𝑡𝑢≤₃𝑘(₁𝑛/𝑢+₂𝑙𝜀−1𝑇)𝑇,,exp−𝑢₃+𝑢₃𝑘₁/𝑢₂𝜀(𝑡−(𝑛−1)𝑇)1−1−𝛿₂exp−𝑢₃+(𝑛𝑢+₃𝑘𝑙₁/−𝑢₂1)𝜀𝑇<𝑇𝑡≤𝑛𝑇,.(2.29)
Therefore, there exists a 𝑇₁>0 such that𝑦(𝑡)≤𝑧(𝑡)≤𝑧^∗(𝑡)+𝜀₁,(2.30) and it follows that 𝑑𝑥(𝑡)𝑑𝑡≥𝑥(𝑡)𝑢^𝑐1−𝑥𝜁³_𝑚−𝑐¹𝜁₃−𝑢¹𝑘₁𝑧^∗𝑥𝑡(𝑡)+𝜀₁,𝑡≠(𝑛+𝑙−1)𝑇,₊=1−𝛿₁𝑥(𝑡),𝑡=(𝑛+𝑙−1)𝑇,(2.31) for 𝑡≥𝑇₁. Let (𝑁+𝑙−1)𝑇≥𝑇₁, integrating (2.31) on ((𝑛+𝑙−1)𝑇,(𝑛+𝑙)𝑇],𝑛≥𝑁, so 𝑥((𝑛+𝑙)𝑇)≥𝑥((𝑛+𝑙−1)𝑇)1−𝛿₁exp^{(𝑛+𝑙)𝑇(𝑛+𝑙−1)𝑇}𝑢^𝑐1−𝑥𝜁³_𝑚−𝑐¹𝜁₃−𝑢¹𝑘₁𝑧^∗=𝑥((𝑛+𝑙−1)𝑇)𝜓,(𝑡)+𝜀₁𝑑𝑡(2.32) then 𝑥((𝑁+𝑛+𝑙)𝑇)≥𝑥((𝑁+𝑙)𝑇)𝜓^𝑛→∞ 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^∗=inf_𝑡>𝑡1{𝑥(𝑡)<𝜁₃}, then there are two possible cases for 𝑡^∗.Case 1. If 𝑡^∗=(𝑛₁+𝑙−1)T,𝑛₁∈𝑁, then 𝑥(𝑡)≥𝜁₃ for 𝑡∈[𝑡₁,𝑡^∗] and (1−𝛿₁)𝜁₃≤𝑥(𝑡^∗+)=(1−𝛿₁)𝑥(𝑡^∗)<𝜁₃, and select 𝑛₂,𝑛₃∈𝑁 such that 𝑛₂−1𝑇>ln𝜀¹/𝑀+𝑝1−𝛿₁−𝑢₃+𝑢₃𝑘₁/𝑢₂𝜁₃,^𝑛2exp𝑛₂𝜓₁𝑇𝜓^𝑛3>1−𝛿₁^𝑛2exp𝑛₂+1𝜓₁𝑇𝜓^𝑛3₁>1,(2.33) where 𝜓₁=𝑢_𝑐(1−(𝜁₃/𝑥_𝑚))−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)=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩1−𝛿²^{𝑛−(𝑛1+1)}𝑧𝑛¹𝑇⁺−𝑝×exp−𝑢₃1−1−𝛿₂exp−𝑢₃+𝑢₃𝑘₁/𝑢₂𝜁₃𝑇+𝑢³𝑘₁𝑢₂𝜁₃𝑡−𝑛¹1−𝛿₂𝑇+𝑧^∗(𝑡),(𝑛−1)𝑇<𝑡≤(𝑛+𝑙−1)𝑇,^𝑛−𝑛1𝑧𝑛¹𝑇⁺−𝑝×exp−𝑢₃1−1−𝛿₂exp−𝑢₃+𝑢₃𝑘₁/𝑢₂𝜁₃𝑇+𝑢³𝑘₁𝑢₂𝜁₃𝑡−𝑛¹𝑇+𝑧^∗(𝑡),(𝑛+𝑙−1)𝑇<𝑡≤𝑛𝑇,(2.34) and 𝑛₁+1≤𝑛≤𝑛₁+𝑛₂+𝑛₃. Therefore, |𝑧(𝑡)−𝑧^∗(𝑡)|<(𝑀+𝑝)exp((−𝑢₃+(𝑢₃𝑘₁/𝑢₂)𝜁₃)(𝑡−𝑛₁𝑇))<𝜀₁ and 𝑦(𝑡)≤𝑧(𝑡)≤𝑧^∗(𝑡)+𝜀₁ for 𝑛₁𝑇+(𝑛₂−1)𝑇≤𝑡≤𝑡^∗+𝑇^ which implies that (2.31) holds for 𝑡^∗+𝑛₂𝑇≤𝑡≤𝑡^∗+𝑇^. So as in Step 1, 𝑥𝑡^∗+𝑇≥𝑥𝑡^∗+𝑛₂𝑇𝜓^𝑛3.(2.35) From system (1.1) 𝑑𝑥(𝑡)𝑑𝑡≥𝑥(𝑡)𝑢^𝑐1−𝑥𝜁³_𝑚−𝑐¹𝜁₃−𝑢¹𝑘₁𝑥𝑡𝑀,𝑡≠(𝑛+𝑙−1)𝑇,₊=1−𝛿₁𝑥(𝑡),𝑡=(𝑛+𝑙−1)𝑇,(2.36) for 𝑡∈[𝑡^∗,𝑡^∗+𝑛₂𝑇]. Integrating (2.36) on 𝑡∈[𝑡^∗,𝑡^∗+𝑛₂𝑇] such that 𝑥𝑡^∗+𝑛₂𝑇≥𝜁₃1−𝛿₁^𝑛2exp𝑛₂𝜓₁𝑇,(2.37) thus, 𝑥(𝑡^∗+𝑇^)≥𝜁₃(1−𝛿₁)^𝑛2exp(𝑛₂𝜓₁𝑇)𝜓^𝑛3>𝜁₃, which is a contraction.
Let 𝑡₂=inf_𝑡>𝑡∗{𝑥(𝑡)>𝜁₃}, then 𝑥(𝑡)≤𝜁₃ when 𝑡∈(𝑡^∗,𝑡₂) and 𝑥(𝑡₂)=𝜁₃. For 𝑡∈(𝑡^∗,𝑡₂),𝑥(𝑡)≥𝜁₃1−𝛿₁^𝑛2+𝑛3exp𝑛₂+𝑛₃𝜓₁𝑇.(2.38) Let 𝜁^1=𝜁₃(1−𝛿₁)^𝑛2+𝑛3exp((𝑛₂+𝑛₃)𝜓₁𝑇), then we have 𝑥(𝑡)≥𝜁^1 for 𝑡∈(𝑡^∗,𝑡₂). For 𝑡>𝑡₂, the same arguments can be continued since 𝑥(𝑡₂)≥𝜁₃.Case 2. If 𝑡^∗≠(𝑛₁+𝑙−1)𝑇,𝑛₁∈𝑁, then 𝑥(𝑡)≥𝜁₃ for 𝑡∈[𝑡₁,𝑡^∗], and 𝑥(𝑡^∗)=𝜁₃; suppose that 𝑡^∗∈((𝑛^1+𝑙−1)𝑇,(𝑛^1+𝑙)𝑇),𝑛₁∈𝑁. There are also two possible cases for 𝑡∈(𝑡^∗,(𝑛^1+𝑙)𝑇).Subcase 1. If 𝑥(𝑡)≤𝜁₃ for all 𝑡∈(𝑡^∗,(𝑛^1+𝑙)𝑇), as in Case 1, we can prove that there must be a 𝑡^1∈[(𝑛^1+𝑙)𝑇,(𝑛^1+𝑙)𝑇+𝑇^] such that 𝑥(𝑡^1)>𝜁₃. Here, we omit it.
Let 𝑡₃=inf_𝑡>𝑡∗{𝑥(𝑡)>𝜁₃}, then 𝑥(𝑡)≤𝜁₃ when 𝑡∈(𝑡^∗,𝑡₃) and 𝑥(𝑡₃)=𝜁₃. For 𝑡∈(𝑡^∗,𝑡₃)𝑥(𝑡)≥𝜁₃1−𝛿₁^𝑛2+𝑛3exp𝑛₂+𝑛₃+1𝜓₁𝑇.(2.39) Let 𝜁₁=𝜁₃(1−𝛿₁)^𝑛2+𝑛3exp((𝑛₂+𝑛₃+1)𝜓₁𝑇)<𝜁^1, 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^∗,(𝑛^1+𝑙)𝑇) such that 𝑥(𝑡)>𝜁₃. Let 𝑡₄=inf_𝑡>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 𝑥(𝑡)≥𝑥𝑡^∗exp𝜓₁𝑡−𝑡^∗≥𝜁₃exp𝜓₁𝑇>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, 𝑢_𝑐=0.5, 𝑥_𝑚=15, 𝑥_ℎ=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<𝑇_max; 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 𝑇_max≈9. If the period of the pulses 𝑇 is larger than 𝑇_max, 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>𝑇_max (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.

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(b)

Figure 1 

Dynamic behavior of system (1.1). When 𝑇<𝑇max≈9, 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.

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(b)

Figure 2 

(a) A period 𝑇 attractor when 𝑇=10, (b) time series algae population when 𝑇=10.

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(b)

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.

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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.

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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 [22–25]. 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.

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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<𝑇_max. 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<𝑇_max. 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/𝑢_𝑐)ln(1/(1−𝛿₁))≈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).