Complex Dynamics of Beddington–DeAngelis-Type Predator-Prey Model with Nonlinear Impulsive Control

According to resource limitation, a more realistic pest management is that the impulsive control actions should be adjusted according to the densities of both pest and natural enemy in the field, which result in nonlinear impulsive control. -erefore, we have proposed a Beddington–DeAngelis interference predator-prey model concerning integrated pest management with both density-dependent pest and natural enemy population. We find that the pest-eradication periodic solution is globally stable if the impulsive period is less than the critical value by Floquet theorem.-e condition of permanent is established, and a stable positive periodic solution appears via a supercritical bifurcation by bifurcation theorem. Finally, in order to investigate the effects of those nonlinear control strategies on the successful pest control, the bifurcation diagrams showed that the model exists with very complex dynamics. Consequently, the resource limitation may result in pest outbreak in complex ways, which means that the pest control strategies should be carefully designed.


Introduction
Since pest outbreak can cause serious economic loss, pest control has been becoming an increasing concern to entomologists and society all over the world. Several pest control strategies can be used for farmers. As well known, chemical pesticide can directly and rapidly kill large proportion of pest, and it is the only way to prevent economic losses in many cases. Biological control is the practice of releasing of natural enemies to control pests [1,2], sometimes which has a highly efficacious and more active role in some pest situations. However, in order to avoid the resistance development of pests to the control tactic and to protect the environment quality, different pest control techniques should be combined together rather than against overuse of a single control strategy. In particular, integrated pest management(IPM) incorporates a variety of cultural, biological, and chemical methods to high efficiency control of the pest populations, which has been proved that it is more efficient long-term strategy for pest control than the classical one (such as biological control or chemical control) [3][4][5].
It is reasonable and accurate that impulsive differential equations mathematically simulate the evolution of biological behaviors and complex biological phenomena, which provide conditions for people to assist in the design of IPM strategies and understand the biological phenomena from a mathematical point of view [1,3,[6][7][8][9]. In recent years, the impulsive differential systems with integrated pest management have been systematically studied and developed rapidly [10,11], which enriched its basic theory and analytical techniques of impulsive differential system [12][13][14][15][16][17][18][19][20]. However, one of the main assumptions in previous literature is that a certain proportion of pest population is killed when the pesticide is applied. Meanwhile, a constant natural enemy is released [21][22][23][24][25][26][27][28], which means that the agricultural resources have almost no effect on IPM.
In reality, the release methods and ratios of numbers of natural enemies will inevitably be affected by the limitation of agricultural resources because of the unbalance development of agricultural, such as agricultural capital, labour forces, biological resources, and pesticides. erefore, the release ratios of numbers of natural enemies according to current density in the field could significantly affect the effect of pest control strategy. In order to take the resource limitation into the IPM strategy, several predator-prey models with nonlinear impulse have been proposed [29][30][31][32] and mainly focused on establishing the global stability conditions. However, the nonlinear impulsive function mentioned above is only related to the density of the natural enemy population in the field. Based on resource limitation, the densities of the pest and natural enemy should be carefully monitored before IPM measures are applied. A more realistic case is that the methods for the instantaneous releasing numbers of natural enemies should be based on the dynamic changes of pest and natural enemy densities. In other words, the higher the number of pest population or the lower the number of predator population in the field, the higher the number of predator population should be released and vice versa, which has not been studied until now. erefore, in order to take the resource limitation into account and to understand how the nonlinear density regulatory factor for the natural enemies affect the dynamics of predator-prey model, we propose the following predatorprey model with Beddington-DeAngelis functional response and nonlinear impulsive control: where x(t) and y(t) are the densities of prey and predator populations, respectively, and all parameters are positive constants. IPM strategy (the nonlinear impulse) is applied at each discrete time point nT. 0 ≤ q 1 , q 2 ≤ 1 present survival rate of prey and predator after harvesting or pesticides; q 2 ≥ 1 means that the pesticides only affect the pest and an impulsive increase of the predator population density is induced by release of predators. Moreover, we choose the nonlinear saturation functions or density-dependent functions as follows: , t � nT, (2) λ 1 , λ 2 ≥ 0 is the maximal release amount of the predator according to the densities of prey and predator populations respectively, and θ 1 , θ 2 ≥ 0 represent the shape parameter. In particular, the system with λ 1 � 0, θ 2 � 0 (i.e., linear impulsive perturbations) has been investigated in [27,28]. We assume that the densities of the natural enemy populations are updated to y(t + ) � q 2 y(t) + (λ 1 x(t)/1 + θ 1 x(t)) +(λ 2 /1 + θ 2 y(t)) at each time point nT, which is a more reasonable control strategy than previous literature [31,32]. e purpose of this paper proposes a Beddington-DeAngelis interference model with nonlinear impulsive control to address how the nonlinear impulsive control actions affect the successful pest control strategies. By using the Floquet theorem and small-amplitude perturbation skills, we obtain that the pest-eradication periodic solution is globally stable if the period of impulsive T is less than a critical value, and a sufficient condition for the permanence of the system is obtained. Moreover, when the trivial periodic solution loses its stability, we obtain that a nontrivial periodic solution appears via a supercritical bifurcation by employing a bifurcation theorem. By bifurcation diagrams, we show that the model presents more rich and interesting dynamic behavior including periodic doubling bifurcation, period-halving bifurcations, chaotic solutions, and multistability. Finally, we give some related biological implications.

Global Stability of the Pest-Eradication Periodic Solution
As we know, eradicating the pest population is an important purpose of IPM strategy, so the existence and global stability of the pest-eradication periodic solution play a crucial role in studying the dynamical behavior. For this, we firstly study the properties of the subsystem Subsystem (3) is a nonlinear growth model, by using the same methods as those in reference [32], and we can have the following result. where erefore, when q 2 exp(− δT) < 1, we have that model (1) has complete expression for pest-eradication periodic solution (x p (t), y p (t)) � (0, y * exp(− δ(t − nT))).
Next, we will present a condition which guarantees the local and global asymptotic stability of pest-eradication periodic solution (x p (t), y p (t)) of model (1).

Theorem 1.
e pest-eradication periodic solution (x p (t), y p (t)) is locally asymptotically stable provided that

Complexity
Furthermore, (x p (t), y p (t)) is globally asymptotically stable provided Proof. e local stability of the pest-eradication solution may be determined by the behavior of small-amplitude perturbations of the solution. Defining and Φ(0) � I is the identity matrix. From the third and fourth equations of (1), one has that erefore, if each eigenvalues of the following matrix has absolute value less than one, then the solution (x p (t), y p (t)) of model (1) is locally stable, and * is not needed to calculate the exact form. Note that all multipliers are It is easy to see that μ 2 ≤ q 2 exp(− δT) < 1. Furthermore, So, we obtain μ 2 ≥ q 2 exp(− δT) − 1 > − 1, which means |μ 2 | < 1. Since T 0 (r − (αy p (t)/c + by p (t)))dt � (1/bδ), ln ((c + by * exp(− δT))/(c + by * )), according to the Floquet theory of impulsive differential equation, the pest-eradication periodic solution (x p (t), y p (t)) is locally asymptotically stable if Next, we will show the global attractivity provided condition (6) is satisfied. From the comparison theorem of impulsive equation, we obtain y(t) ≥ y p (t) − ε for all t large enough. Also, it is easy to see that x(t) < K + ε for all t large enough.
For simplicity, we may assume that y(t) ≥ y p (t) − ε and x(t) < K + ε for all t ≥ 0. If condition (6) holds true, then we choose an ε > 0 such that From model (1), we obtain Integrating on (nT, (n + 1)T], one obtains Complexity 3 us, x(nT) � x(0 + )η n and consequently x(nT) ⟶ 0 as n ⟶ ∞. erefore, Following, we only need to prove y(t) ⟶ y p (t) as t ⟶ ∞. ere must exist a 0 < ε 1 < (δc/β) and from which we can have the following equation: By Lemma 1, model (17) has a globally asymptotically stable periodic solution According to the comparison theorem, we get y p (t) ≤ y(t) ≤ z(t) and z(t) ⟶ z p (t), z p (t) ⟶ y p (t) as t ⟶ ∞. Hence, for any ε 1 > 0, we have for T 2 ≥ T 1 > 0. Furthermore, let ε ⟶ 0, we get y p (t) − ε 1 < y(t) < y p (t) + ε 1 for t large enough. In other words, y(t) ⟶ y p (t) as t ⟶ ∞ for t large enough. e proof is completed.

Permanence
Persistence is an important property of dynamical systems for addressing the long-term survival of all components of a system. Now, we investigate the sufficient condition for the permanence of model (1).
Proof. Suppose that (x(t), y(t)) is a solution of (1) with x(0) > 0, y(0) > 0. It is easy to know that x(t) < M, y(t) < M for all t > 0, M > (rc/α). Define m 2 � y * exp(− δT) − ε. From eorem 1, it is easy to see that y(t) ≥ m 2 for t large enough. en, we shall find an m 1 such that x(t) ≥ m 1 for all t that are large enough. We will do it in the following two steps.
If not, we can define since the continuity of x(t) and x(t * ) � m 3 . We only need to consider two possible cases.
Case (2a): x(t) < m 3 for t ∈ (t * , (n 1 ′ + 1)T]. In this case, we continue this process by using step case (1), we can prove that there we obtain 3 . In this case, a similar argument as above, erefore, integrating equation (26) on [t * , t)(t ≤ t)), we can get that x(t) ≥ x(t * )exp(ρ(t − t * )) ≥ m 1 . us, the similar argument can be continued for both cases since x(t) ≥ m 1 for some t > t 1 . is completes the proof.

Bifurcation
Now, we will deal with the existence of nontrivial solution near the pest-eradication solution. We use the bifurcation theory in earlier publications [33]. Complexity Theorem 3. Model (1) has a positive nontrivial periodic solution when T � T * , which is supercritical if q 2 > θ 2 λ 2 and Kα < 4rbc.
Proof. It is convenient for the computation to exchange x and y and change the period T to τ.

Numerical Simulation
To confirm our theoretical results and facilitate their interpretation, we will focus on the complex dynamics by bifurcation analysis numerically, which can obtain the properties of a dynamics system.
Firstly, we investigate the effect of pulse period T on dynamical of system. Figure 1 shows that model (1) could exist with complex and interesting dynamic behavior with increase of parameter T, such as period-doubling bifurcation, period-halving bifurcations, chaos band, and nonunique dynamics, i.e., several attractors may coexist with the same T. For example, Figure 2 indicates that two different attractors can coexist with each other with the same T � 7. If we choose different initial value (x 0 , y 0 ) � (1.4, 2.1) and (x 0 , y 0 ) � (1.8, 2.1), a 3T-periodic solution coexists with Tperiodic solution, which indicates that the final stable states of pest and natural enemy population depend on their initial densities. All these results confirm that varying impulsive period T could dramatically change the dynamics of model (1).
It follows from Figure 3 that the nonlinear impulsive parameter λ 1 affects the dynamics of model (1). As parameter λ 1 increases, system (1) experiences period-doubling bifurcation, chaotic, period-halving bifurcations, and multiple stability. When λ 1 � 9.4, two attractors with different amplitudes appear, i.e., a T-periodic solution and 2T-periodic solution coexist (see Figure 4). erefore, the initial values of both the pest and predator populations are crucial.
e above results reveal that the parameter λ 1 can dramatically change dynamics of system (1).
From the bifurcation diagrams Figure 5, we observe that a positive periodic solution appears when the pest-eradication solution loses its stability. e behavior of positive periodic solution is kept until q 2 ≈ 3.9, and then a perioddoubling bifurcation occurs, which means that a T-periodic solution disappears suddenly at this point and 2T-periodic solution appears. With the increasing of q 2 , a series of period-doubling bifurcations lead model (1) from periodicity to chaos. When q 2 ≥ 5.16 and nearby 5.16, the chaos disappears and a 3T-periodic solution appears. As q 2 increases further, the evidence for 3T-periodic solution leading to chaos can be seen. Following these, the system displays a series of period-halving bifurcations.
Similarly, we investigate the effect of parameter q 1 on dynamic of system. e bifurcation diagrams with respect to parameter q 1 in the range [0, 0.8] are shown in Figure 6. We can observe that model (1) also displays very complex dynamical behaviors with q 1 increasing.

Conclusion
Based on resource limitation, the optimal pest control strategy is that the instantaneous releasing numbers of natural enemies should be adjusted according to the densities of both pest and natural enemy in the field. A more natural understanding is that when the higher the number of pest population or the lower the number of predator population in the field, the higher the number of predator population should be released and the converse is also true. For this, we have investigated effects of nonlinear impulsive perturbations on a predator-prey model with Beddington-DeAngelis functional response. We have proven that there is a global stability of pest-eradication periodic solution if the impulsive period T < T * by using the Floquet theorem and small amplitude perturbation skills, and model (1) is permanent when the period T > T * . Hence, T � T * plays a bifurcation threshold, and the system bifurcates to a positive periodic solution via supercritical bifurcation once a threshold condition is reached. By bifurcation diagrams, we can show that the system contains very rich dynamical behavior, including period-doubling bifurcation, periodhalving bifurcations, chaos, and nonunique attractors, i.e., the system could exist with two stable positive periodic solutions and even more complex dynamics (see Figures 2  and 4). Also, bifurcation analyses reveal that the final dynamics of the system depends on the initial densities, and the nonlinear impulsive may result in complexity of pest control. All those results confirm that the pest control strategy should be carefully designed once the nonlinear impulsive control measures have been taken into account.
Based on the present study, we found that the system with nonlinear impulsive control actions provides more rich results and more realistic than the previous systems with linear impulsive control, and thus nonlinear impulsive control should be taken into account when implementing integrated pest management. However, the aim of IPM should reduce pest populations to below the economic threshold rather than eradication, which can be naturally and accurately described by the state-dependent impulsive differential equations and result in more difficulty for analyzing the global dynamics. We leave for future research.

Data Availability
ere were no data used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.