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

Authors: Changtong Li; Xiaozhou Feng; Yuzhen Wang; Xiaomin Wang
Journal: Complexity
 (2020)

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## Abstract

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. Therefore, 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. The 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.


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## Body

## 1. 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) [3–5].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, 6–9]. 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 [12–20]. 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 [21–28], 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. Therefore, 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 [29–32] 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.Therefore, 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 predator-prey model with Beddington–DeAngelis functional response and nonlinear impulsive control:(1)dxtdt=rxt1−xtK−αxtytc+xt+bytdytdt=βxtytc+xt+byt−δyt,t≠nT,xt+=q1xtyt+=q2yt+λ1xt1+θ1xt+λ21+θ2yt,t=nT.where xtandyt 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≤q1,q2≤1 present survival rate of prey and predator after harvesting or pesticides; q2≥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:(2)yt+=q2yt+λ1xt1+θ1xt+λ21+θ2yt,t=nT,λ1,λ2≥0 is the maximal release amount of the predator according to the densities of prey and predator populations respectively, and θ1,θ2≥0 represent the shape parameter. In particular, the system with λ1=0,θ2=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 yt+=q2yt+λ1xt/1+θ1xt+λ2/1+θ2yt at each time point nT, which is a more reasonable control strategy than previous literature [31, 32].The 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 impulsiveT 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.


## 2. 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(3)dytdt=−δyt,t≠nT,yt+=q2yt+λ21+θ2yt,t=nT.Subsystem (3) is a nonlinear growth model, by using the same methods as those in reference [32], and we can have the following result.Lemma 1.
When1−q2exp−δT>0, model (3) has a globally stable periodic solution:(4)ypty∗exp−δt−nT,t∈nT,n+1T,where y∗=A+1−1/2θ2exp−δT with A=4λ2θ2exp−δT/1−q2exp−δT is a positive constant.Therefore, whenq2exp−δT<1, we have that model (1) has complete expression for pest-eradication periodic solution xpt,ypt=0,y∗exp−δt−nT.Next, we will present a condition which guarantees the local and global asymptotic stability of pest-eradication periodic solutionxpt,ypt of model (1).Theorem 1.
The pest-eradication periodic solutionxpt,ypt is locally asymptotically stable provided that(5)T<1rln1q1−αrbδlnc+by∗exp−δTc+by∗.
Furthermore,xpt,ypt is globally asymptotically stable provided(6)T<1rln1q1−αrbδlnc+K+by∗exp−δTc+K+by∗.Proof.
The local stability of the pest-eradication solution may be determined by the behavior of small-amplitude perturbations of the solution. Definingut=xt−xpt,vt=yt−ypt, then the fundamental matrix Φt of model (1) satisfies(7)dΦtdt=r−αyptc+bypt0βyptc+bypt−δΦt,and Φ0=I is the identity matrix. From the third and fourth equations of (1), one has that(8)unT+vnT+=q10λ1q2−λ2θ21+θ2ypT2unTvnT=BTunTvnT.
Therefore, if each eigenvalues of the following matrix(9)M=BTΦT=q1exp∫0Tr−αyptc+byptdt0∗q2−λ2θ21+θ2ypT2exp−δThas absolute value less than one, then the solution xpt,ypt of model (1) is locally stable, and ∗ is not needed to calculate the exact form. Note that all multipliers are(10)μ1=q1exp∫0Tr−αyptc+byptdt,μ2=q2−λ2θ21+θ2ypT2exp−δT.
It is easy to see thatμ2≤q2exp−δT<1. Furthermore,(11)λ2θ2exp−δT1+θ2ypt2≤4λ2θ2exp−δT1+4λ2θ2exp−δT+12<1.
So, we obtainμ2≥q2exp−δT−1>−1, which means μ2<1. Since ∫0Tr−αypt/c+byptdt=1/bδ,lnc+by∗exp−δT/c+by∗, according to the Floquet theory of impulsive differential equation, the pest-eradication periodic solution xpt,ypt is locally asymptotically stable if(12)T<1rln1q1−αrbδlnc+by∗exp−δTc+by∗.
Next, we will show the global attractivity provided condition (6) is satisfied. From the comparison theorem of impulsive equation, we obtain yt≥ypt−ε for all t large enough. Also, it is easy to see that xt<K+ε for all t large enough.
For simplicity, we may assume thatyt≥ypt−ε and xt<K+ε for all t≥0. If condition (6) holds true, then we choose an ε>0 such that(13)η≜q1exp∫0Tr−αypt−εc+K+ε+bypt−εdt<1.
From model (1), we obtain(14)dxtdt≤xtr−αypt−εc+K+ε+bypt−ε,t≠nT,xt+=q1xt,t=nT.
Integrating onnT,n+1T, one obtains(15)xn+1T=q1xnTexp∫nTn+1Tr−αypt−εc+K+ε+bypt−εdt=xnTη.
Thus,xnT=x0+ηn and consequently xnT⟶0 as n⟶∞. Therefore, xt⟶0 as n⟶∞, since 0<xt≤q1xnTexprT for t∈nT,n+1T.
Following, we only need to proveyt⟶ypt as t⟶∞. There must exist a 0<ε1<δc/β and T1>0 such that 0<xt<ε1 for t>T1. Again, for simplicity, it is assumed that 0<xt<ε1 holds true for t≥0. Then, we deduce that λ1xt/1+θ1xt<ε1λ1 and(16)−δyt≤dydt≤ytβε1c−δ,from which we can have the following equation:(17)dztdt=βε1c−δzt=−δ1zt,t≠nT,zt+=q2zt+ε1λ1+λ21+θ2zt,t=nT.
By Lemma1, model (17) has a globally asymptotically stable periodic solution zpt=z∗exp−δ1t−nT, where z∗=−A1+A12+4B1ε1λ1+λ2/2B1 with A1=1−q2+ε1λ1θ2exp−δ1T,B1=θ21−q2exp−δ1Texp−δ1T and t∈nT,n+1T.
According to the comparison theorem, we getypt≤yt≤zt and zt⟶zpt,zpt⟶ypt as t⟶∞. Hence, for any ε1>0, we have(18)ypt−ε1<yt<zpt+ε1,for T2≥T1>0. Furthermore, let ε⟶0, we get ypt−ε1<yt<ypt+ε1 for t large enough. In other words, yt⟶ypt as t⟶∞ for t large enough. The proof is completed.


## 3. 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).Theorem 2.
Model (1) is permanent if T>1/rln1/q1−α/rbδlnc+by∗exp−δT/c+by∗ holds true.Proof.
Suppose thatxt,yt is a solution of (1) with x0>0,y0>0. It is easy to know that xt<M,yt<M for all t>0, M>rc/α. Define m2=y∗exp−δT−ε. From Theorem 1, it is easy to see that yt≥m2 for t large enough. Then, we shall find an m1 such that xt≥m1 for all t that are large enough. We will do it in the following two steps.Step 1. Let m3>0,ε1>0 be small enough such that δ2=δ−βm3/c+m3>0 and η≜q1exprT−rm3T/K−α/bδ2lnc+bu∗exp−δ2T/c+bu∗−αε1T/c>1, where u∗=−A2+A22+4B2m3λ1+λ2/2B2 with A2=1−q2+m3λ1θ2exp−δ2T,B2=θ21−q2exp−δ2Texp−δ2T. We will prove that xt<m3 cannot hold for all t>0. Otherwise,(19)dytdt≤−δ2yt.From Lemma1, we then obtain yt≤ut and ut⟶upt,t⟶∞, where ut is the solution of(20)dutdt=−δ2ut,t≠nT,ut+≤q2ut+λ1m3+λ21+θ2ut,t=nT,andupt=u∗exp−δ2t−nT,t∈nT,n+1T with u∗=−A2+A22+4B2m3λ1+λ2/2B2.Consequently, there exists aT^>0 such that yt≤ut<upt+ε1 and(21)dxtdt≥xtr−rm3K−αytc+byt≥xtr−rm3K−αupt+αε1c+bupt+bε1≥xtr−rm3K−αuptc+bupt−αε1c,fort>T^. Furthermore, we get(22)dxtdt≥xtr−rm3K−αuptc+bupt−αε1c,t≠nT,xt+=q1xt,t=nT,fort>T^. Integrating (22) on nT,n+1T, where nT>T^, we obtain(23)xn+1T≥xnT+exp∫nTn+1Tr−rm3K−αuptc+bupt−αε1cdt=xnTη.Then,xn+kT>xnTηn⟶∞ as k⟶∞, which is a contradiction. Therefore, there exists a t1>0 such that xt1≥m3.Step 2. If xt≥m3 for all t≥t1, then model (1) is permanent. If not, we can define t∗=inft>t1xt<m3, Then, xt≥m3 for t∈t1,t∗ since the continuity of xt and xt∗=m3. We only need to consider two possible cases.Case (1):t∗=n1T. For some n1∈Z+, then we have q1m3≤xt∗+=q1xt∗<m3. Select n2,n3∈Z+ such that(24)q1n2ηn3expn2+1η1T>1,whereρ=r−m3r/K−α/cM<0. Let T′=n2T+n3T, we will show that there exists a t∗<t2≤t∗+T′ such that xt2≥m3. If not, by (20) with ut∗+=yt∗+, we can see that yt≤ut≤upt+ε1 for t∈t∗+n2T,t∗+T′. An argument similar to Step 1 yields(25)xt∗+T′≥xt∗+n2Tηn3.Sinceyt≤M, when t∈t∗,t∗+n2T, we get(26)dxtdt≥xtr−m3rK−αcM,t≠nT,xt+=q1xt,t=nT.Furthermore, we integrate equation (26) on t∗,t∗+n2T, then we know(27)xt∗+n2T≥m3q1n2−1expn2ρT.From above, we obtainxt∗+T′≥m3q1n2−1expn2ρTηn3>m3 which leads to a contradiction.Now, lett˜=inft>t∗xt≥m3, then xt≤m3 for t∈t∗,t˜ and xt˜=m3. For any t∗<t<t˜, let us assume that there exists k∈Z+ such that t∈t∗+k−1T,t∗+kT and k≤n2+n3, so, from (26), we obtain(28)xt≥xt∗+q1k−1expk−1ρTexpρt−t∗+k−1T≥m3q1kexpkρT≥m3q1n2+n3expn2+n3ρT.Letm1′=m3q1n2+n3expn2+n3η1T, thus, for t∈t∗,t˜, we get xt≥m1′. For t>t˜, we can continue the same arguments since xt˜≥m3.Case (2):t∗≠n1T,n1∈Z+. We have xt∗=m3 since xt is continuous. Suppose t∗∈n1′T,n1′+1T,n1′∈Z+, we consider the following two cases for t∈t∗,n1′+1T.Case (2a):xt<m3 for t∈t∗,n1′+1T. In this case, we continue this process by using step case (1), we can prove that there exists a t^=inft>t∗xt≥m3 such that xt<m3,t∈t∗,t^, and xt^=m3. For any t∈t∗,t^, suppose t∈n1′T+l′−1T,n1′T+l′T,l′≤1+n2+n3, we obtain(29)xt≥m3q1l′−1expl′η1T≥m3q1n2+n3expn2+n3+1ρT≜m1.Sincem1<m1′, thus, for t∈t∗,t^, we have xt≥m1.Case (2b): there is at′∈t∗,n1′+1T such that xt′≥m3. In this case, a similar argument as above, there is t¯=inft>t∗xt>m3 such that xt<m3,t∈t∗,t¯. Therefore, integrating equation (26) on t∗,tt≤t¯, we can get that xt≥xt∗expρt−t∗≥m1. Thus, the similar argument can be continued for both cases since xt≥m1 for some t>t1. This completes the proof.Remark 1.
DefineT∗=1/rln1/q1−α/rbδlnc+by∗exp−δT/c+by∗, the pest-eradication periodic solution loses its stability if T>T∗. Therefore, T∗ is the critical threshold value to discriminate between stability and permanence.


## 4. 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].Theorem 3.
Model (1) has a positive nontrivial periodic solution when T=T∗, which is supercritical if q2>θ2λ2 and Kα<4rbc.Proof.
It is convenient for the computation to exchangex and y and change the period T to τ.(30)dxtdt=βxtytc+yt+bxt−δxdytdt=ry1−yK−αxtytc+yt+bxt,t≠nτ,xt+=q2xt+λ1yt1+θ1yt+λ21+θ2xtyt+=q1yt,t=nτ.
LetΦ be the solution of the pulse-free system associated with system (30). Also, we get Xt=Φt,X0 with X0=X0. We define the mapping Θ1,Θ2:R2⟶R2 by(31)Θ1x,y=q2x+λ1y1+θ1y+λ21+θ2x,Θ2x,y=q1y,ξt=ypt,0,and the map F1,F2:R2⟶R2 by(32)F1x,y=βxtytc+yt+bxt−δx,F2x,y=ry1−yK−αxtytc+yt+bxt.
Furthermore, to establish the nontrivial periodic solution of system (30), we employ the techniques in [33]. Then, we have(33)∂Φ1t,X0∂x=exp∫0t∂F1ξt∂xdt,∂Φ1t,X0∂y=∫0texp∫st∂F1ξs∂yds∂F1ξv∂xexp∫0v∂F2ξs∂ydsdv,∂Φ2t,X0∂x=0,∂Φ2t,X0∂y=exp∫0t∂F2ξt∂ydt>0.
Thus, by simple calculations, we obtain(34)d0′=1−∂θ2∂y⋅∂Φ2∂yτ0,X0=1−q1exp∫0τ0r−αyptc+byptdt,where τ0 is the root of d0=0. Actually, it is easy to see that d0′=0 is equivalent to τ0=T∗.
Also, we obtain that(35)a0′=1−∂θ1∂x·∂Φ1∂xτ0,X0=1−q1exp−DT>0,b0′=−∂θ1∂x⋅∂Φ1∂y+∂θ1∂y·∂Φ2∂yτ0,X0=−q2−θ2λ21+θ2y∗exp−DT2∂Φ1τ0,X0∂y−λ1∂Φ2τ0,X0∂y.
Note thatb0′<0 if q2>θ2λ2.
Following, we should calculate the second-order partial derivatives:(36)∂2Φ2τ0,X0∂x∂y=∫0τ0exp∫sτ0∂F2ζt∂ydt∂2F2ζt∂x∂yexp∫0s∂F2ζt∂ydtds=−∫0τ0exp∫st∂F2ζt∂ydtbαypsc+byps3exp∫0s∂F2ζt∂ydtds<0.
By the same methods as shown above, we have(37)∂2Φ2τ0,X0∂y2=∫0τ0exp∫sτ0∂F2ζt∂ydt∂2F2ζt∂y2exp∫0s∂F2ζt∂ydtds+∫0τ0exp∫sτ0∂F2ζt∂ydt∂2F2ζs∂y∂x⋅∫0sexp∫vs∂F1ζt∂xdt∂F1ζs∂yexp∫0v∂F2ζt∂ydtdv=∫0τ0exp∫sτ0∂F2ζt∂ydt−2rK+2αypsc+byps2exp∫0s∂F2ζt∂ydtds−∫0τ0exp∫sτ0∂F2ζt∂ydtbαypsc+byps3·∫0sexp∫vs∂F1ζt∂xdtαypvc+bypvexp∫0v∂F2ζt∂ydtdvds,∂2Φ2τ0,X0∂τ∂y=∂F2ζτ0∂yexp∫0τ0∂F2ζt∂ydt=r−αypτ0c+bypτ0exp∫0τ0r−αyptc+byptdt.
Therefore, we can deduce that(38)B=−∂2Θ2∂x∂y∂Φ1τ0,X0∂τ+∂Φ1τ0,X0∂x1a0′∂Θ1∂x∂Φ1τ0,X0∂τ∂Φ2τ0,X0∂y−∂Θ2∂y∂2Φ2τ0,X0∂x∂y1a0′∂Θ1∂x∂Φ1τ0,X0∂τ+∂2Φ2τ0,X0∂τ∂y=−q1∂2Φ2τ0,X0∂x∂y1a0′q2−θ2λ21+θ2ypτ02yp′τ0+r−αypτ0c+bypτ0exp∫0τ0r−αyptc+byptdt,C=−2∂2Θ2∂x∂y−b0′a0′∂Φ1τ0,X0∂x+∂Φ1τ0,X0∂y∂Φ2τ0,X0∂y−∂2Θ2∂y2∂Φ2τ0,X0∂y2+2a0′b0′∂Θ2∂y∂2Φ2∂x∂y−∂Θ2∂y∂2Φ2τ0,X0∂y2=2q1b0′a0′∂2Φ2τ0,X0∂x∂y−q1∂2Φ2τ0,X0∂y2.
Now, we determine the sign ofB, For this, let gt=r−αypt/c+bypt, then g′t=αδcy∗exp−δt/c+bypt2>0, so gt is strictly increasing. Since ∫0τ0gtdt=ln1/q1>0, which indicates that gτ0>0, i.e. ∂2Φ2τ0,X0/∂y∂τ>0. Therefore, B<0. Defining φt=−r/K+αt/c+bt2, it is easy to know that −r/K+αt/c+bt2≤0 if Kα<4rbc, which means φypt=−r/K+αypt/c+bypt2≤0 for all ypt>0. So, we have ∂2Φ2τ0,X0/∂y2<0. From above, we have C>0.
Hence,BC<0; according to Theorem 2 of [33], a supercritical bifurcation occurs at T=T∗.Figure 1
Bifurcation diagrams of model (1) with respect to T. The parameter values are as follows. r=2.72,K=10,c=1,b=0.001,α=1.2,β=0.48,δ=0.59,θ1=12,θ2=1,q1=0.62,q2=0.98,λ1=4.4,andλ2=10.2.
(a)(b)


## 5. 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 periodT 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 x0,y0=1.4,2.1 and x0,y0=1.8,2.1, a 3T-periodic solution coexists with T-periodic 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).Figure 2
Two coexisting attractors of model (1) with T=7. The other parameter values are identical to those in Figure 1. The initial conditions are as follows: A−Bx0,y0=1.4,2.1;C−Dx0,y0=1.8,2.6.
(a)(b)(c)(d)It follows from Figure3 that the nonlinear impulsive parameter λ1 affects the dynamics of model (1). As parameter λ1 increases, system (1) experiences period-doubling bifurcation, chaotic, period-halving bifurcations, and multiple stability. When λ1=9.4, two attractors with different amplitudes appear, i.e., a T-periodic solution and 2T-periodic solution coexist (see Figure 4). Therefore, the initial values of both the pest and predator populations are crucial. The above results reveal that the parameter λ1 can dramatically change dynamics of system (1).Figure 3
Bifurcation diagrams of model (1) with respect to λ1. The other parameters are identical to those in Figure 1 and K=10.5,T=10.
(a)(b)Figure 4
Two coexisting attractors of model (1) with λ1=9.4. The other parameter values identical to those in Figure 3. The initial conditions are as follows: A−Bx0,y0=1.4,2.1;C−Dx0,y0=1.8,2.6.
(a)(b)(c)(d)From the bifurcation diagrams Figure5, we observe that a positive periodic solution appears when the pest-eradication solution loses its stability. The behavior of positive periodic solution is kept until q2≈3.9, and then a period-doubling bifurcation occurs, which means that a T-periodic solution disappears suddenly at this point and 2T-periodic solution appears. With the increasing of q2, a series of period-doubling bifurcations lead model (1) from periodicity to chaos. When q2≥5.16 and nearby 5.16, the chaos disappears and a 3T-periodic solution appears. As q2 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.Figure 5
Bifurcation diagrams of model (1) with respect to q2. Parameter values are as follows: r=2.12,K=10,c=1,b=0.01,α=1.2,β=0.48,δ=0.39,θ1=12.5,θ2=1.5,q1=0.11,λ1=1.5,andλ2=2.
(a)(b)Similarly, we investigate the effect of parameterq1 on dynamic of system. The bifurcation diagrams with respect to parameter q1 in the range 0,0.8 are shown in Figure 6. We can observe that model (1) also displays very complex dynamical behaviors with q1 increasing.Figure 6
Bifurcation diagrams of model (1) with respect to q2. Parameter values are as follows: r=2.18,K=5.5,c=1.15,b=0.01,α=0.55,β=0.87,δ=0.34,θ1=1.2,θ2=2,q2=0.98,λ1=1.4,λ2=2.2,andT=16.5.
(a)(b)


## 6. 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 periodT<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, period-halving 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.



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*Source: 8829235-2020-12-11.xml*