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

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.


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 time t nT ; the system can be described as follows: 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; u 1 x t / x t k 1 is one of the most well-known functional responses describing a prey-predator interaction, called Holling-Type II functional response; c 1 is the intraspecific competition rate of the algae; u 3 is the average mortality rate for fish; 0 ≤ δ 1 , δ 2 ≤ 1 represent the fraction of the algae and fish which die due to the harvesting or chemical poisoning at t 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.

Preliminaries and Mathematical Analysis
is the map defined by the right-hand sides of the first and second equations of system 1.
the upper right derivative of V t, X with respect to the impulsive differential system 1.1 is defined as Remark 2.2. 1 The solution of system 1.1 is a piecewise continuous function with X : R → R 2 , 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 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:

2.2
System 2.2 is a periodically forced linear system, then we get that is a positive periodic solution of system 2.2 with the initial values since the general solution of 2.2 is

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 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 and as a result, where Φ t satisfies and Φ 0 I, the identity matrix.The linearization of the third and fourth equations of 2.2 becomes The linearization of fifth and sixth equations of 2.2 becomes The stability of the periodic solution 0, y * t is determined by the eigenvalues of Therefore, all eigenvalues of θ are given by

2.13
According to Floquet theory, 0, y * t is locally asymptotically stable if λ 1 < 1, that is to say, In the following, we prove the global attractivity.Choose a ε > 0 such that 2.17 which leads to

2.18
Hence, x n l T ≤ x lT ξ n 1 and x n l T → 0 as n → ∞.Therefore, Next, we prove that y t → y * t as t → ∞.For 0 < ε < u 3 k 1 /u 2 , 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 , From Lemma 2.5 and comparison theorem of impulsive equation, z 1 t ≤ y t ≤ z 2 t and z 1 t → y * t , z 2 t → y * t as t → ∞, where z 1 t and z 2 t are solutions of

2.23
Let ε → 0 such that 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 .
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 − δ 2 exp −u 3 T / 1 − 1 − δ 2 exp −u 3 T − ε ≈ ζ 2 for t large enough.Therefore, it is only necessary to find an ζ 1 > 0 such that x t ≥ ζ 1 for t large enough.We prove this in the following two steps. Step then it is easy to prove that x t < ζ 3 cannot hold for all t.Otherwise,

2.27
Then, y t ≤ z t and z t → z * t t → ∞ , where z t is the solution of

2.29
Therefore, there exists a T 1 > 0 such that and it follows that it is a contradiction because x t is ultimately bounded.Therefore, there exists a t 1 > 0 such that x t 1 ≥ ζ 3 .
Step 2. If x t ≥ ζ 3 for all t > t 1 , then the proof will be complete.Otherwise, let t * inf t>t 1 {x t < ζ 3 }, then there are two possible cases for t * .

2.34
and

2.35
From system 1.1 For t > t 2 , the same arguments can be continued since There are also two possible cases for t ∈ t * , n 1 l T .Subcase 1.If x t ≤ ζ 3 for all t ∈ t * , n 1 l T , as in Case 1, we can prove that there must be a Here, we omit it.
Let  When T < T max ≈ 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.
Since x t 4 ≥ ζ 3 for t > t 4 , the same arguments can be continued.Therefore, x t > ζ 1 for t > t 1 , so system 1.1 is permanent.The proof is complete.

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: u 1 0.175, u 2 0.3, u 3 0.18, u c 0.5, x m 15, x h 20, k 1 0.6, c 1 0.05, δ 1 0.4, δ 2 0.4, p 4, L 0.07, x 0 0.5, and y 0 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.
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 perioddoubling 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.

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 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 − δ 1 ≈ 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 δ 1 0, δ 2 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.

Figure 1 :
Figure1: Dynamic behavior of system 1.1 .When T < T max ≈ 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.

Figure 2 :
Figure 2: a A period T attractor when T 10, b time series algae population when T 10.

Figure 3 :
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 .

Figure 5 :
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.

Figure 6 :
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 .
is claimed here that there must be a t 2 ∈ t * , t * T such that x t 2 > ζ 3 .Otherwise, considering 2.28 with z t * y t * , it follows that t 3 , and when t > t 3 , the same arguments can be got since x t 3 ≥ ζ 3 .There exists a t ∈ t * , n 1 l T such that x t > ζ 3 .Let t 4 inf t>t * {x t > ζ 3 } such that x t ≤ ζ 3 when t ∈ t * , t 4 and x t 4 ζ 3 .When t ∈ t * , t 4 , the inequality 2.36 holds.Integrating 2.36 on t ∈ t * , t 4 , then x t ≥ x t * exp ψ 1 t − t * ≥ ζ 3 exp ψ 1 T > ζ 1 .