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.
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 . 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 : where and are the respective densities of prey and predator at time , the constant is the intrinsic growth rate of the prey population, is the coefficient of intraspecies competition, is the per-capita rate of predation of the predator, is the half-saturation constant, which depicts a critical concentration of the nutrient composition to maintain the normal growth of the predator, is the death rate of the predator, and is the rate of conversion of a consumed prey to a predator.
More recently, the author of  has developed the following three-species Holling type IV system by introducing spraying of pesticides and periodic constant release of mid-level predators at different fixed times: where is the per-capita rate of predation of the top predator, is the half-saturation constant, is the death rate of the top predator, and is the rate of conversion of consumed mid-level predators to the top predator.
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: where , , and are the respective densities of the prey, top predator, and mid-level predator at time , is the per-capita rate of predation of the top predator, is the half-saturation constant, and is the rate of conversion of consumed prey to the top predator. The meanings of the other parameters are the same as in (1.1) and (1.2). , , and , , represent the fractions of prey and predator which die because of harvesting or for other reasons, is the magnitude of immigration or stocking of the predator, and 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.
2. Mathematical Analysis
First, some useful notations and statements will be provided for use in subsequent proofs. The following definitions will be useful.
Let . Denote by the map defined by the right-hand sides of the first, second, third, and fourth equations of system (1.3). Let; then is continuous on , exists, and is locally Lipschitzian in .
Definition 2.1. Let ; then for , the upper right derivative of with respect to the impulsive differential system (1.3) can be defined as The solution of system (1.3) is a piecewise continuous function , is continuous on , and exists. Obviously, the smoothness properties of guarantee the global existence and uniqueness of a solution of system (1.3); for details, see [25, 26].
Lemma 2.3. Assume that is a solution of system (1.3) with ; then for all . Furthermore, , if .
Lemma 2.4 (see ). Let and assume that where is continuous on for , exists, and is nondecreasing. Let be a maximal solution of the scalar impulsive differential equation: existing on . Then implies that , where is any solution of system (1.3). If certain smoothness conditions on can be established to guarantee the existence and uniqueness of solutions for (2.3), then is exactly a unique solution of (2.3).
Now consider a special case of Lemma 2.4. Let denote the class of real piecewise continuous functions defined on .
Lemma 2.5 (see ). Let and let it satisfy the inequalities: where and , and u0 are constants, and is a strictly increasing sequence of positive real numbers. Then, for ,
For convenience, some basic properties can be defined for the following subsystems of system (1.3): Then the following lemma results.
Next, some main theorems will be proposed.
Theorem 2.7 (boundedness). There exists a constant such that , , and for each solution of system (1.3) for all sufficiently large .
Proof. Let be a solution of system (1.3); let . Then From the first part, it is known that , so . Choosing , then . The following can then be obtained: From Lemma 2.5, When , (2.11) . It is clear that is bounded for sufficiently large . Therefore, , , and 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 is locally asymptotically stable if and , where , , , and .
Proof. The local stability of the periodic solution may be determined by considering the behavior of small-amplitude perturbations of the solution. Define Substituting (2.12) into system (1.3), it is possible to obtain a linearization of the system as follows: which can be written as where the satisfies with , where is the identity matrix and Hence, the stability of the periodic solution is determined by the eigenvalues of If the absolute values of all eigenvalues are less than one, the periodic solution is locally stable. Then all eigenvalues of can be denoted by , and : Therefore, where , , and . Clearly, and only if , and only if . According to the Floquet theory of impulsive differential equations, the periodic solution is locally stable. This completes the proof.
Theorem 2.9. The solution is said to be globally stable if .
Proof. By Theorem 2.7, there exists a constant such that , , for each solution of system (1.3) for all sufficiently large . From the first equation in system (1.3), it is possible to obtain Then, and using (2.21), Denoting as: , by Lemmas 2.4 and 2.6, there exists a , and it is possible to select . If is small enough, then for all and satisfying . Therefore, (2.20) ; if , then (2.20) . As , ; this implies that there exist values of and such that for , and when , these satisfy . From the second equation in system (1.3), . Let be the solution of the following equation: From Lemmas 2.4 and 2.6, and the third equation in system (1.3), where is the periodic solution of (2.23): Integrating (2.24) over , where and When , as . For , which implies that as . Therefore, it can be assumed that for . From the second equation in system (1.3), . Let and be solutions of the following equation: From Lemmas 2.4 and 2.6, and , as . Also note that and as and . Therefore, as . This completes the proof.
Theorem 2.10. System (1.3) is permanent if
Proof. Let be a solution of system (1.3). From Theorem 2.7, there exists a constant such that , , for each solution of system (1.3) for all sufficiently large . Let ; then , , , and .
By Lemmas 2.4 and 2.6, , where Then, , for sufficiently large . Therefore, it is necessary only to find an such that and for sufficiently large . This can be done in the following two steps.
Step 1. Choose small enough that when (2.30) and (2.11) hold, where This step will show that and for some . Assuming the contrary, the following system results: By Lemma 2.4, , , and . By Lemma 2.6, , ,. Thus, Integrating (2.35) over yields Therefore, and .
Therefore, and as . This implies that and as , which contradicts the boundedness of and .
Step 2. If and for all , then the proof is complete. If not, let ; then and for and ,. By Step 1, there exists a such that . Set ; then and for , and . 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 , such that and , . Let . If , there must exist a subsequence such that as . From Step 1, this can lead to a contradiction with the boundedness of and . Therefore, . Then, Let ; then , and . This completes the proof.
3. Numerical Analysis
To study the dynamics of system (1.3), a period was chosen, and the impulsive control parameter 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 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 will be examined. A time series of the system response is shown in Figure 2. It is apparent that the solution , which is said to be globally stable, is in agreement with Theorem 2.9. The bifurcation diagrams are shown in Figure 3.
Figures 1 and 3 illustrate the complex dynamics of system (1.3), such as period-doubling 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 , the dynamics of the system are not obvious, but as increases, the dynamics become more evident. The system enters into periodic windows with a chaotic band, as shown in Figures 4(a) (not obviously chaotic) and 4(b) (-periodic solution). When is between 0.414 and 2.689, chaos is intense, as shown in Figure 4(c). When moves beyond 2.689, the chaotic behavior disappears, and periodic windows appear, as shown in Figures 4(d) (4-periodic solution) and Figure 4(e) (2-periodic solution). When , the chaotic attractor increases in strength, and chaos reappears, as shown in Figure 4(f). For greater than 3.076, periodic windows appear, as shown in Figure 4(g). For in the interval 3.258 to 3.486, chaos reemerges, as shown in Figure 4(h). As the value of increases further, the system enters into a stable state, as shown in Figure 4(i).
3.2. 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 . 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.
3.3. Strange Attractors and Power Spectra
To understand the qualitative nature of strange attractors, power spectra can be used . 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.
4. 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 and the impulsive control parameter . 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 cannot be persistent although the mid-level predator is persistent. This is because the mid-level predator is easy to escape the top predator , 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.
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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