Dynamic Analysis of an Impulsively Controlled Predator-Prey Model with Holling Type IV Functional Response

The dynamic behavior of a predator-prey model with Holling type IV functional response is investigated with respect to impulsive control strategies. The model is analyzed to obtain the conditions under which the system is locally asymptotically stable and permanent. Existence of a positive periodic solution of the system and the boundedness of the system is also confirmed. Furthermore, numerical analysis is used to discover the influence of impulsive perturbations. The system is found to exhibit rich dynamics such as symmetry-breaking pitchfork bifurcation, chaos, and nonunique dynamics.


Introduction
In recent years, impulsive control strategies in predator-prey models have become a major field of inquiry.Many authors have studied the dynamics of predator-prey models with impulsive control strategies 1-10 .Much research has also been done on three-species food chain systems with impulsive perturbations 11-17 .Holling-type functional responses are well known, and therefore many authors have studied the dynamics of predator-prey models with different Holling-type functional responses with respect to an impulsive control strategy.For example, the dynamics of a predator-prey model with Holling type I functional response with respect to an impulsive control strategy have been reported in 18 .The dynamics of a predator-prey model with Holling type II functional response with an impulsive control strategy were presented in 19, 20 .In 21, 22 , the results of studies of the dynamics of a predator-prey model with Holling type VI functional response with respect to an impulsive control strategy were presented.The following predator-prey model with Holling type VI functional response with respect to an impulsive control strategy was proposed in 23 : where x t and y t are the respective densities of prey and predator at time t, the constant a is the intrinsic growth rate of the prey population, b is the coefficient of intraspecies competition, c 1 is the per-capita rate of predation of the predator, e 1 is the half-saturation constant, which depicts a critical concentration of the nutrient composition to maintain the normal growth of the predator, d 1 is the death rate of the predator, and c 2 is the rate of conversion of a consumed prey to a predator.More recently, the author of 24 has developed the following three-species Holling type IV system by introducing spraying of pesticides and periodic constant release of midlevel predators at different fixed times: Assume that the top predator also eats the prey, or in other words, that the relationship between the top predator and the mid-level predator is not only that of predator and prey, but also that of competitors.To represent this, a predator-prey model with Holling type IV functional response with respect to an impulsive control strategy can be constructed as follows:  , represent the fractions of prey and predator which die because of harvesting or for other reasons, p is the magnitude of immigration or stocking of the predator, and T is the period of impulsive immigration or stocking of the predator.Here it is assumed that the rate of conversion of consumed prey to predator is smaller than the per-capita rate of predation of the predator.
The rest of this paper is organized as follows.Section 2 presents a mathematical analysis of the model.Section 3 describes some numerical simulations.In the last section, a brief discussion is provided, and conclusions are drawn.

Mathematical Analysis
First, some useful notations and statements will be provided for use in subsequent proofs.The following definitions will be useful.
Let R 0, ∞ , R 3 {X x t , y t , z t ∈ R 3 | X ≥ 0}.Denote by f f 1 , f 2 , f 3 the map defined by the right-hand sides of the first, second, third, and fourth equations of system 1.3 .LetV 0 {V : V nT , X exists, and V is locally Lipschitzian in X.
Definition 2.1.Let V ∈ V 0 ; then for nT, n 1 T × R 3 , the upper right derivative of V t, X with respect to the impulsive differential system 1.3 can be defined as The solution of system 1.3 is a piecewise continuous function X : R ×R 3 , X t is continuous on nT, n 1 T , n ∈ N, and X nT lim t → nT X T exists.Obviously, the smoothness properties of f guarantee the global existence and uniqueness of a solution of system 1.3 ; for details, see 25, 26 .Definition 2.2.System 1.3 is permanent if there exists M ≥ m > 0 such that, for any solution x t , y t , z t of system 1.3 with a positive initial value Lemma 2.3.Assume that X t is a solution of system 1.3 with X 0 ≥ 0; then X t ≥ 0 for all t ≥ 0. Furthermore, X t > 0, t > 0 if X 0 > 0.
Lemma 2.4 see 23 .Let V ∈ V 0 and assume that where f, h ∈ PC 1 R , R and α k ≥ 0, β k and u 0 are constants, and τ k k ≥ 0 is a strictly increasing sequence of positive real numbers.Then, for t > 0,

2.5
For convenience, some basic properties can be defined for the following subsystems of system 1.3 :

2.6
Then the following lemma results.
Lemma 2.6.For a positive periodic solution y * t of system 2.6 and a solution y t of system 2.6 with initial value y 0 y 0 ≥ 0, where

2.8
Next, some main theorems will be proposed.

Theorem 2.7 boundedness .
There exists a constant M such that x t ≤ M, y t ≤ M, and z t ≤ M for each solution X x t , y t , z t of system 1.3 for all sufficiently large t.
Proof.Let x t , y t , z t be a solution of system 1.3 ; let u t x t y t z t , t ≥ 0 .Then

2.9
From the first part, it is known that The following can then be obtained:

2.10
From Lemma 2.5, It is clear that u t is bounded for sufficiently large t.Therefore, x t , y t , and z t are bounded.This completes the proof.
Next the stability of a prey and top-predator eradication periodic solution will be examined.
Theorem 2.8.The solution 0, y * t , 0 is locally asymptotically stable if Proof.The local stability of the periodic solution 0, y * t , 0, 0 may be determined by considering the behavior of small-amplitude perturbations of the solution.Define which can be written as where the φ t satisfies with φ 0 I, where I is the identity matrix and Hence, the stability of the periodic solution 0, y * t , 0 is determined by the eigenvalues of If the absolute values of all eigenvalues are less than one, the periodic solution 0, y * t , 0 is locally stable.Then all eigenvalues of φ can be denoted by λ 1 , λ 2 , and λ 3 :

2.18
Therefore, where According to the Floquet theory of impulsive differential equations, the periodic solution 0, y * t , 0 is locally stable.This completes the proof.
Theorem 2.9.The solution 0, y * t , 0 is said to be globally stable if Proof.By Theorem 2.7, there exists a constant M > 0 such that x t ≤ M, y t ≤ M, z t ≤ M for each solution X x t , y t , z t of system 1.3 for all sufficiently large t.From the first equation in system 1.3 , it is possible to obtain Denoting λ as: by Lemmas 2.4 and 2.6, there exists a t 1 > 0, and it is possible to select ε > 0. If ε is small enough, then y t ≥ y * 1 t − ε for all t ≥ t 1 and satisfying a < c 1 λ/ 1 e 1 M 2 .Therefore, 2.20 , then 2.20 < 0. As t → ∞, x t → 0; this implies that there exist values of ε 1 and T 1 such that x t < ε 1 for t ≥ T 1 , and when Let y 1 t be the solution of the following equation:

2.23
From Lemmas 2.4 and 2.6, and the third equation in system 1.3 , where y * 1 t is the periodic solution of 2.23 : Integrating 2.24 over nT, n 1 T , where which implies that z t → 0 as t → ∞.Therefore, it can be assumed that z t < ε 2 for t > 0.
From the second equation in system 1.

2.30
Proof.Let x t , y t , z t be a solution of system 1.3 .From Theorem 2.7, there exists a constant M > 0 such that x t ≤ M, y t ≤ M, z t ≤ M for each solution X x t , y t , z t of system 1.3 for all sufficiently large t.Let M 0 max{M, M/c 3 }; then x t ≤ M 0 , y t ≤ M 0 , z t ≤ M 0 , and y t ≥ − d 1 M 0 y t .By Lemmas 2.4 and 2.6, y t ≥ u * t − ε, ε > 0 , where for sufficiently large t.Therefore, it is necessary only to find an m 2 > 0 such that x t ≥ m 2 and z t ≥ m 2 for sufficiently large t.This can be done in the following two steps.Step 1. Choose m 1 > 0, ε 1 > 0 small enough that when 2.30 and 2.11 hold, where

2.33
This step will show that x t 1 ≥ m 1 and z t 1 ≥ m 1 for some t 1 > 0. Assuming the contrary, the following system results:

2.35
Integrating 2.35 over nT, n 1 T yields

2.36
Therefore, Therefore, σ k → ∞ and ρ k → ∞ as k → ∞.This implies that x t → ∞ and z t → ∞ as t → ∞, which contradicts the boundedness of x t and z t .
Step 2. If x t ≥ m 1 and z t ≥ m 1 for all t ≥ t 1 , then the proof is complete.If not, let t * inf t≥t 1 {x t < m 1 , z t < m 1 }; then x t ≥ m 1 and z t ≥ m 1 for t ∈ t 1 , t * and x t * m 1 ,z t * m 1 .By Step 1, there exists a t > t * such that z t ≥ m 1 .Set t 2 inf t>t * {x t ≥ m 1 , z t ≥ m 1 }; then x t < m 1 and z t < m 1 for t ∈ t * , t 2 , and z t 2 m 1 .This process can be continued by repeating Step 1.If this process stops after a finite number of repetitions, the proof is complete.If not, there exists an interval sequence t n , t n 1 , n ∈ N , such that x t ≥ m 1 and a subsequence Step 1, this can lead to a contradiction with the boundedness of x t and z t .Therefore, T < ∞.Then,

Bifurcation
To study the dynamics of system 1.3 , a period T was chosen, and the impulsive control parameter p was defined as the bifurcation parameter.The bifurcation diagram provides a summary of the essential dynamic behavior of the system 27-29 .
First, the influence of the period T will be investigated.The bifurcation diagrams are shown in Figure 1.The results are in accordance with Theorem 2.8.Next, the influence of the impulsive control parameter p will be examined.A time series of the system response is shown in Figure 2. It is apparent that the solution 0, y * t , 0 , which is said to be globally stable, is in agreement with Theorem 2.9.The bifurcation diagrams are shown in Figure 3.
To see the dynamics of system 1.3 clearly, a phase diagram with a different value of the parameter p corresponding to the bifurcation diagrams in Figure 3 is shown in Figure 4.
Figures 1 and 3 illustrate the complex dynamics of system 1.3 , such as perioddoubling cascades, symmetry-breaking pitchfork bifurcations, chaos, and nonunique dynamics.Because all the bifurcation diagrams are similar, only one of them will be explained.Part a of Figure 3 will be taken as an example.When p ∈ 0, 0.414 , the dynamics of the

The Largest Lyapunov Exponent
The largest Lyapunov exponent is always calculated to detect whether a system is exhibiting chaotic behavior.The largest Lyapunov exponent takes into account the average exponential rates of divergence or convergence of nearby orbits in phase space 30 .A positive largest Lyapunov exponent indicates that the system is chaotic.If the largest Lyapunov exponent is negative, then periodic windows or a stable state must exist.Using the largest Lyapunov exponent, it is possible to see when the system is chaotic, at what time periodic windows disappear, and when the system is stable.The largest Lyapunov exponents corresponding to Figures 1 and 3 were calculated and are plotted in Figures 5 and 6.

Strange Attractors and Power Spectra
To understand the qualitative nature of strange attractors, power spectra can be used 31 .From the discussion in Section 3.2, it is known that the largest Lyapunov exponent for strange attractor a is 0.475 and that for strange attractor b is 0.0144.This means that these are both chaotic attractors, and the fact that the exponent of a is larger than that of b means that the chaotic dynamics of a are more intense than those of b .Considering the power spectra of these attractors, the spectrum of strange attractor b is composed of strong broadband components and sharp peaks, as shown in Figure 7 d .By contrast, in the spectrum of the strong chaotic attractor a , it is not easy to distinguish any sharp peaks, as can be seen in Figure 7 c .By means of power spectra, it is possible to determine that b originates in a strong limit cycle, but that a comes from a set of weak limit cycles.

Conclusions and Remarks
In this paper, the dynamic behavior of a predator-prey model with Holling type IV functional response with respect to an impulsive control strategy has been investigated.The conditions for locally asymptotically stable and globally stable periodic solutions and for system permanence have been determined.It has been determined that an impulsive control strategy changes the dynamic behavior of the model.Complex dynamic patterns also have been observed in continuous-time predator-prey or three-species food-chain models 32-34 .Numerical simulations were performed to obtain bifurcation diagrams with respect to the period T and the impulsive control parameter p.Using computer-based simulation, phase diagrams were generated to reveal details of the bifurcation behavior, such as period-doubling cascades, symmetry-breaking pitchfork bifurcations, chaos, and nonunique dynamics.The largest Lyapunov exponents were also simulated to verify the chaotic dynamics of the system.In addition, power spectra were used to understand the qualitative nature of strange attractors.On the basic of numerical simulations, the main difference of this paper from other papers is that the top predator z cannot be persistent although the mid-level predator y is persistent.This is because the mid-level predator y is easy to escape the top predator z, which is a variable to the competitive.It can be concluded that an impulsive control strategy is an effective method to control the system dynamics of a predator-prey model.

Figure 4 :
Figure 4: Periodic behavior and chaos corresponding to Figure 3: a phase diagram for p 0.05, b phase diagram for p 0.5, c phase diagram for p 2, d phase diagram for p 2.75, e phase diagram for p 2.85, f phase diagram for p 3.05, g phase diagram for p 3.15, h phase diagram for p 3.4, and i phase diagram for p 4.3.

Figure 7 :
Figure 7: Strange attractors and power spectra: a strange attractor when p 2.2, b strange attractor when p 3.039, c power spectrum of attractor a , d power spectrum of attractor b .
3is the per-capita rate of predation of the top predator, e 2 is the half-saturation constant, d 2 is the death rate of the top predator, and c 4 is the rate of conversion of consumed mid-level predators to the top predator.