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Dynamic Analysis of a Predator-Prey (Pest) Model with Disease in Prey and Involving an Impulsive Control Strategy
The dynamic behaviors of a predator-prey (pest) model with disease in prey and involving an impulsive control strategy to release infected prey at fixed times are investigated for the purpose of integrated pest management. Mathematical theoretical works have been pursuing the investigation of the local asymptotical stability and global attractivity for the semitrivial periodic solution and population persistent, which depicts the threshold expression of some critical parameters for carrying out integrated pest management. Numerical analysis indicates that the impulsive control strategy has a strong effect on the dynamical complexity and population persistent using bifurcation diagrams and power spectra diagrams. These results show that if the release amount of infective prey can satisfy some critical conditions, then all biological populations will coexist. All these results are expected to be of use in the study of the dynamic complexity of ecosystems.
Predator-prey models with disease are a major concern and are now becoming a new field of study known as ecoepidemiology. The disease factor in predator-prey systems has been firstly considered by Anderson and May . In subsequent years, many authors studied the dynamics of ecological models with infected prey, and their papers mainly focused on this issue [2–8]. The infection rate and the predation rate are the two primary factors, which can control the chaotic dynamics of an ecoepidemiological system . Das et al.  studied the HP model  by introducing disease in prey populations, which can be described as follows: where , and are respectively the susceptible prey population, infected prey population, the intermediate predator population, and the top-predator population, and are the maximal predation rate of intermediate predator for susceptible and infected prey, respectively, is the maximal predation rate of top-predator for intermediate predator, and are the half saturation constant for functional response of intermediate and the top-predator respectively, is the conversion rate of susceptible prey to intermediate predator, is the conversion rate of infected prey to intermediate predator, and is the conversion rate of intermediate predator to top predator.
Through the dimensionless transformation (seeing ), the system can change into the following form:
In recent decades, technological revolutions have recently hit the industrial world; thus, infected population can now be controlled by many methods such as spraying pesticides and vaccination. It is well known that pest management involves using pesticides and releasing natural enemies, which have been focused by many researchers [12–14]. Control of an infected population can be achieved by chemical or biological control or both, which is called an impulsive control strategy in biomathematics. Systems with impulsive control strategies to describe time-varying processes are characterized by the fact that at certain moments, their states undergo abrupt change. Recently, impulsive control strategies have been recently introduced into population ecology [15–18], chemotherapeutic approaches to treat disease , and food webs [20–25].
Based on the two aspects discussed, the authors constructed a predator-prey model with disease in prey (a pest) and involving an impulsive control strategy for the purpose of integrated pest management. The impulsive control strategy was used to introduce infected prey (a pest) at a fixed time on the basis of system (1.2). The predator-prey model with disease in prey and involving an impulsive control strategy can be described by the following differential equations: where , and are respectively the densities of susceptible prey (a pest), infected prey (a pest), the intermediate predator (natural enemy), and the top predator at time . Then, , , , and . We have where and are the maximal predation rates of the intermediate predator on susceptible and infected prey respectively; is the maximal predation rate of the top predator on the intermediate predator; and are the half-saturation constants for functional response of the intermediate prey and the top predator respectively; is the conversion rate of susceptible prey to intermediate predators; and are, respectively, the conversion rate of infected prey to intermediate predators and the conversion rate of the intermediate predator to the top predator; , , are the death rates of infected prey, the intermediate predator, and the top predator, respectively; is the incidence rate; is the intrinsic growth rate; is the carrying capacity (see ); is the introduced amount of infective prey population at , , , where is the period of the impulsive control. It is known that pest outbreak will cause some serious ecological and economic problems, and we can directly gather infected prey to increase the amount of infected prey and indirectly carry out integrated pest management.
The paper is organized as follows: in the next section, a mathematical analysis of the model is carried out. Section 3 describes some numerical simulations, and the last section contains a brief discussion.
2. Mathematical Analysis
Some important notations, lemmas, and definitions will be provided, which are frequently used in subsequent proofs.
Let . Denote as the map defined by the right-hand side of the first, second, third, and fourth equations of system (1.3). Let, then is said to belong to class if(1) is continuous on , and for each exists;(2) is locally Lipschitzian in .
The solution of system (1.3) is a piecewise continuous function , where is continuous on, and exists. Obviously the smoothness properties of can guarantee the global existence and uniqueness of the solution of system (1.3); for details see [26–28].
Lemma 2.6 (see ). Let , and assume that where is continuous in for , existing, and nondecreasing. Let be a maximal solution of the scalar impulsive differential equation as follows: existing on . Then , implying that , where is any solution of system (1.3). Note that if certain smoothness conditions on exist to guarantee the existence and uniqueness of solutions for (2.4), then is the unique solution of (2.4).
For convenience, some basic properties of certain subsystems of system (1.3) are now provided as follows: Therefore, the following lemma holds.
Lemma 2.7 (see ). For a positive periodic solution of system (2.5) and the solution of system (2.5) with initial value , , , where Next, the stability of susceptible prey and of predator-eradication periodic solutions will be studied.
Theorem 2.8. The solution is said to be locally asymptotically stable if .
Proof. The local stability of periodic solution may be determined by considering the behavior of small-amplitude perturbations of the solution. Define
Substituting (2.7) into (1.3), a linearization of the system can be obtained as follows:
This can be rewritten as
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 , where , , , .
Clearly, with only if according to the Floquet theory of impulsive differential equations, and the periodic solution is locally stable. This completes the proof.
Theorem 2.9. The solution is said to be globally attractive if and
Proof. Let ; then
By Lemma 2.5, there exists a constant such that , , , for each solution of system (1.3) with sufficiently large .
Then, By Lemmas 2.6 and 2.7, there exists a , and an can be selected to be small enough so that for all . By (2.15), Let and . Therefore, when , by (2.16), . So and , , as . It is known from the fact that the limiting state of system (1.3) is exactly system (2.5) and from Lemma 2.7 that is globally attractive. This completes the proof.
Theorem 2.10. System (1.3) is permanent if , ,
Proof. From Lemma 2.5, there exists a constant such that , , , for each solution of system (1.3) with sufficiently large. From (2.15), it is known that for large enough .
Therefore, it is only necessary to find a that satisfies , , . This will be achieved in the following two steps.
Let , and .
Then First, it will be proved that there exists a such that , , and because is ultimately bounded.
Next, it will be proved that , , cannot hold for all .
Otherwise, Then let be the solution of It follows that and where .
So there exists a such that Then According to the above conditions, ; then and , , as ; however, this is a contradiction. Therefore, is ultimately bounded.
Second, if , , for all , then the objective has been attained. To show this, let , and it follows that for and that . Suppose that . Select such that > , , where Let . It is claimed that there must be a such that . Otherwise, consider (2.21) with . Then For , , it can be shown that , for , and ≥ . For , it can be shown that ≥ ; then, ≥ , so ≥ > , which is a contradiction. Therefore, there exists a such that , resulting in ≥ + + + .
When , the same procedure can be performed. According to the above discussion, if , every solution of system (1.3) will eventually enter and remain in the region . This completes the proof.
3. Numerical Analysis
To study the dynamics of system (1.3), the period and the impulsive control parameter are used as the bifurcation parameter. The bifurcation diagram provides a summary of the basic dynamic behavior of the system [30, 31].
First, the influence of the period is studied using the time series shown in Figure 1. The bifurcation diagrams are shown in Figures 2 and 3. Next, the influence of the impulsive control parameter is investigated. The bifurcation diagrams for this are shown in Figure 4.
To clearly see the dynamics of system (1.3), it is necessary to examine the phase diagrams at different value of the period and parameter corresponding to the bifurcation diagrams in Figures 2 and 4; the results of this analysis are shown in Figures 5 and 6.
Figures 2, 3, and 4 reveal the complex dynamics of system (1.3), including period-doubling cascades, symmetry-breaking pitchfork bifurcation, chaos, and nonunique dynamics. Because every bifurcation diagram is similar, only one needs to be explained. Take Figure 4(a) as an example. When , the dynamics of the system are not obvious, but with increasing , the dynamics become more obvious. The system enters into a chaotic band with periodic windows. When is between 0.124 and 0.153, the chaotic behavior is intense, as can be seen in Figure 6(a). When moves beyond 0.153, the chaotic behavior disappears. When , the chaotic attractor gains in strength, and the chaotic behavior appears again. When becomes greater than 0.219, periodic windows appear, as can be seen in Figures 6(b) and 6(c). When is in the interval between 0.328 and 0.35, chaotic behavior ensues, as can be seen in Figure 6(d). As the value of increases further, the system enters a stable state, as is shown in Figures 6(e), 6(f), and 6(g). When moves beyond 1.001, an unexpectedly chaotic phase appears, as is shown in Figure 6(h). It is clear that seasonal disturbances have little effect on the maximum density of all species; however, serious periodic oscillations are generated, and weak periodic solutions lose their stability and move into chaos. In summary, the key factor in the long-term dynamic behavior of system (1.3) is impulse perturbations, but seasonal disturbances can aggravate periodic oscillations and promote the emergence of chaos. Based on the above numerical simulation analysis, it is clear that impulsive control strategy has an important effect on the dynamical behaviors of the system, and weak periodic solutions lose their stability and move into chaos. In summary, the key factor in the long-term dynamic behavior of system (1.3) is impulsive control strategy, but disease disturbances can aggravate periodic oscillations and promote the emergence of chaos.
3.2. The Iargest Lyapunov Exponent
To detect whether the system exhibits chaotic behavior, one of the commonest methods is to calculate the largest Lyapunov exponent. 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, there must be periodic windows or a stable state. Through the largest Lyapunov exponent, it is possible to judge that at what time the system is chaotic, and at what time the system is stable. The largest Lyapunov exponents corresponding to Figures 2, 3, and 4 can be calculated and are shown in Figures 7, 8, and 9, which shows the accuracy and effectiveness of numerical simulation. Moreover, using the simulation of the largest Lyapunov exponents, the existence of chaotic behavior in system (1.3) can be further confirmed.
3.3. Strange Attractors and Power Spectra
To understand the qualitative nature of strange attractors, power spectra are used . From Section 3.2, it is known that the largest Lyapunov exponent for strange attractor (a) is 0.0413, and for strange attractor (b) is 0.124. Therefore, they are both chaotic attractors, and the exponent of (b) is larger than that of (a), which means that the chaotic dynamics of (b) are more extreme than those of (a). The power spectrum of strange attractor (a) is composed of strong broadband components and sharp peaks, as are shown in Figure 10(c). On the contrary, in the spectrum of strong chaotic attractor (b), it is difficult to distinguish any sharp peaks, as can be seen in Figure 10(d). These power spectra can be interpreted as meaning that (a) comes from a strong limit cycle, but that (b) experiences some weak limit cycles. Hence, it is obvious that the impulsive control strategy has a strong effect on the dynamical behaviors of system (1.3) with the period of the impulsive control varying but that (b) experiences some weak limit cycles.
4. Conclusions and Remarks
In the paper, the dynamic behaviors of a predator-prey (pest) model with disease in prey and involving an impulsive control strategy are presented analytically and numerically. The critical conditions are obtained to ensure the local asymptotical stability and global attractivity of semitrivial periodic solution as well as population permanence. Numerical analysis indicates that the impulsive control strategy has a strong effect on the dynamical complexity and population persistent using bifurcation diagrams and power spectra diagrams. In addition, the largest Lyapunov exponents are computed. This computation further confirms the existence of chaotic behavior and the accuracy of numerical simulation. These results revealed that the introduction of disease and the use of an impulsive control strategy can change the dynamic behaviors of the system. The same results also have been observed in continuous-time models of predator-prey or three-species food-chain models [34–37] and other systems . In a word, it should be stressed that the impulsive control strategy is an effective method to control complex dynamics of predator-prey (pest) 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 (NSFC no. 31170338 and no. 30970305) and also by the Key Program of Zhejiang Provincial Natural Science Foundation of China (Grant no. LZ12C03001).
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