# Analysis of an Impulsive One-Predator and Two-Prey System with Stage-Structure and Generalized Functional Response

Authors: Xiangmin Ma; Yuanfu Shao; Zhen Wang; Xianjia Fang; Zhenguo Luo
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

An impulsive one-predator and two-prey system with stage-structure and generalized functional response is proposed and analyzed. By reasonable assumption and theoretical analysis, we obtain conditions for the existence and global attractivity of the predator-extinction periodic solution. Sufficient conditions for the permanence of this system are established via impulsive differential comparison theorem. Furthermore, abundant results of numerical simulations are given by choosing two different and concrete functional responses, which indicate that impulsive effects, stage-structure, and functional responses are vital to the dynamical properties of this system. Finally, the biological meanings of the main results and some control strategies are given.


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## Body

## 1. Introduction and Model Formulation

In real world, the properties of one-predator and one-prey system have been studied widely and many valuable results have been obtained. If examining the cases that there are two preys for a predator, then the above system cannot reflect the real behaviors of individuals accurately, so scholars proposed three-species predator-prey system. The relationship between species in three-species system may take many forms, such as one prey and two predators [1], a food chain [2, 3], or two preys and one predator [4, 5]. On the other hand, for predator-prey model, in description of the relationship between predator and prey, a crucial element is the classic definition of a predator’s functional response. Recently, the dynamics of predator-prey systems with different kinds of functional responses have been studied in relevant literature, such as Holling type [6], Crowley-Martin type [7–9], Beddington-DeAngelis type [10, 11], Watt type [12, 13], and Ivlev type [14]. For example, Gakkhar and Naji [15] investigated the dynamical behaviors of the following three-species system with nonlinear functional response:(1)x1′t=x1tr1-a11x1t-α1x1tyta1+b1x1t+b1x2t,x2′t=x2tr2-a22x2t-α2x2tyta1+b1x1t+b1x2t,y′t=-d+m1α1x1t+m2α2x2ta1+b1x1t+b1x2tyt,where x1(t) and x2(t) represent the two preys densities, respectively, and y(t) represents the density of predators depending on the two preys.However, as Pei et al. [16] pointed out that system (1) could not provide an effective approach because there was no impulsive spraying pesticides or harvesting pest at different fixed moment. We know that pests may bring disastrous effects to their existing system when their amount reaches a certain level. For preventing large economic loss, chemical pesticides are often used in the process of pest management. As a matter of fact, the control on pests often makes pests reduce instantaneously in a short time. In the modeling process, these perturbations are often assumed to be in the form of impulses. Based on traditional models, impulsive differential equations are proposed and extensively used in some applied fields, especially in population dynamics; see [17–19]. The theory of impulsive differential equation is now being recognized richer than the corresponding differential equation without impulses, which plays a key role in the development of biomathematics; see monographs [20, 21] and references cited therein.On the other hand, the stage-structure for predator was also not considered in system (1). In real world, many species go through two or more life stages when they proceed from birth to death. For many animals, their babies are raised by their parents or are dependent on the nutrition from the eggs they stay in. The babies are too weak to produce babies or capture their prey; hence their competition with other individuals of the community can be ignored. Therefore, it is reasonable to introduce stage-structure into competitive or predator-prey models. Many researchers have incorporated it into biological models, where stage-structure is modeled by using a time delay [22–24]. Authors [5] pointed out that when the system contained time delay, it had more interesting behaviors. Their results showed that time delay could cause a stable equilibrium to become unstable and Hopf bifurcation could occur as the time delay crossed some critical values. These obtained results have shown that stage-structure plays a vital role in predator-prey models and stage-structured systems exhibit complicated properties. Moreover, Xu [25] showed that an important factor in modeling of predator-prey is the choice of functional response. Model with generalized functional response exhibited many universal properties, which could be applied to many fields because of its flexibility. Shao and Li [26] considered a predator-prey system with generalized functional response. Their results indicated that generalized functional response caused dynamical behaviors of the system to be very complex.Based on these backgrounds, in this paper, developing system (1) with stage-structure, generalized function response, and impulsive spraying pesticides, we will consider the following one-predator and two-prey system:(2)
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,where x1(t) and x2(t) represent the densities of two different preys, respectively, and we assume that there is no competition between the two preys. y1(t) and y2(t) denote the densities of immature predator and mature predator, respectively. ai is the natural growth rate of xi(t)    (i=1,2). bi and r are coefficients of internal competition of prey xi(t) (i=1,2) and mature predator y2(t), respectively. βi is capture rate of mature predator for xi(t) (i=1,2). λi represents the conversion rate of two preys into reproduction of mature predator. d1 and d2 are death rates of immature predator and mature predator, respectively. τ is the mean length of juvenile period of predator. The term e-dτ denotes the mature rate of immature predator. Function fi(xi(t)) (i=1,2) is adult predator’s functional response. pi (0≤pi<1,  i=1,2) is partial impulsive harvesting of prey by catching or pesticides at moment t=nT  (n=1,2,…).By use of impulsive differential equation theory and some analysis techniques, we aim to investigate the existence and global attractivity of predator-extinction periodic solution and the permanence of (2). Further, by numerical analysis, we try to find out the effects of impulsive and stage-structure on this system.Sincey1(t) does not appear in the first, the second, and the fourth equation of system (2), we can simplify (2) and restrict our attention to the following system:(3)x1′t=x1ta1-b1x1t-β1f1x1ty2t,x2′t=x2ta2-b2x2t-β2f2x2ty2t,y2′t=e-d1τλ1β1f1x1t-τ+λ2β2f2x2t-τy2t-τ-d2y2t-ry22t,t≠nT,x1t+=1-p1x1t,x2t+=1-p2x2t,y2t+=y2t,t=nT,with initial conditions:(4)x1s,x2s,y2s=φ1s,φ2s,φ3s∈C-τ,0,R+3,φi0>0,i=1,2,3.From biological point of view, without loss of generality, in this paper, we assumed thatfi(x)  (i=1,2) is strictly increasing, differential with fi(0)=0, satisfying 0<fi(x)/x<Li (a constant) for all x>0. Further, we only consider (3) in the following biological meaning region:(5)D=x1t,x2t,y1t,y2t∣x1t,x2t,y1t,y2t≥0.The rest of this paper is organized as follows. In Section2, we give some notations, definitions, and lemmas. By using lemmas and impulsive comparison theorem, we discuss the existence of predator-extinction solution and permanence of system (3) in Sections 3 and 4, respectively. In Section 5, numerical simulations are given to show the complicated dynamical behaviors of (3). Finally, we end this paper by a brief discussion in Section 6.


## 2. Preliminaries

In this section, some definitions and lemmas are introduced which are useful for our main results. Solution of (3), denoted by x(t)=(x1(t),x2(t),y1(t),y2(t))T, is piecewise and continuous function: x:R+→R+4,  x(t) is continuous on (nT,(n+1)T],  n∈Z+, and x(nT+)=limt→nT+⁡x(t) exists. Obviously, the global existence and uniqueness of solution of (3) are guaranteed by the smoothness of f=(f1,f2,f3,f4), where f denotes the mapping defined by right side of system (2). For more details refer to [20, 21].Lemma 1 (see [27]).
Consider the following differential equation:(6)g′t=a1gt-τ-a2gt-a3g2t,where ai  (i=1,2,3) is a positive constant and g(t)>0,  t∈[-τ,0]; then we have the following:(i)
ifa1>a2, then limt→∞⁡g(t)=(a1-a2)/a3;(ii)
ifa1<a2, then limt→∞⁡g(t)=0.Lemma 2 (see [10]).
Consider the following impulsive system:(7)x′t=xta-bxt,t≠nT,xt+=1-pxt,t=nT,where a>0, b>0, and p>0. If p<1-e-aT, then system (7) has a positive periodic solution x∗(t) and for any solution x(t) of system (7), we have x(t)-x∗(t)→0, as t→+∞, where(8)x∗t=a1-p-e-aTb1-p-e-aT+bpe-at-nT,x∗0+=a1-p-e-aTb1-e-aT,nT<t≤n+1T.By Lemma 2, we can easily know that if the following hypotheses (H1) and (H2) hold,(H1):
p1<1-e-a1T,(H2):
p2<1-e-a2T.Then (3) has a mature predator-extinction periodic solution (x1∗(t),x2∗(t),0) for t∈(nT,(n+1)T], and for any solution (x1(t),x2(t),y2(t)) of system (3), we have x1(t)→x1∗(t), x2(t)→x2∗(t), and y2(t)→0, t→∞, where(9)xi∗t=ai1-pi-e-aiTbi1-pi-e-aiT+bipie-ait-nT,i=1,2,t∈nT,n+1T,with(10)xi∗0+=ai1-pi-e-aiTbi1-e-aiT,i=1,2.


## 3. Global Attractivity of the Predator-Extinction Periodic Solution

In this section, we investigate the global attractivity of predator-extinction periodic solution(x1∗(t),x2∗(t),0) of system (3).Theorem 3.
Predator-extinction periodic solution(x1∗(t),x2∗(t),0) of system (3) is globally attractive if (H1), (H2), and(H3):
e-d1τ(λ1β1f1(η1)+λ2β2f2(η2))<d2hold, where η1 and η2 are defined in (15) and (16), respectively.Proof.
Since (H3) holds andf1(x), f2(x) are differential for all x>0, we can choose two positive constants ε1 and ε2 to be sufficiently small such that(11)e-d1τλ1β1f1η1+ε1+λ2β2f2η2+ε2<d2.From the first equation of system (3), we have x1′(t)≤x1(t)(a1-b1x1(t)).
Consider the following impulsive comparison system:(12)u1′t=u1ta1-b1u1t,t≠nT,u1t+=1-p1u1t,t=nT.In view of Lemma 2, we obtain that(13)u1∗t=a11-p1-e-a1Tb11-p1-e-a1T+b1p1e-a1t-nT=x1∗t,t∈nT,n+1T,with(14)u1∗0+=a11-p1-e-a1Tb11-e-a1T=x1∗0+,which is unique and globally asymptotically stable positive periodic solution of (12). By use of comparison theorem of impulsive differential equation, there exists k1∈N such that, for the sufficiently small constant ε1 and all t∈(nT,(n+1)T] (n>k1), we have(15)x1t≤x1∗t+ε1≤a11-p1-e-a1Tb11-p1-e-a1T+b1p1e-a1T+ε1≙η1+ε1.Similarly, there exists k2∈N such that, for the sufficiently small constant ε2 and all t∈(nT,(n+1)T] (n>k2), we have(16)x2t≤x2∗t+ε2≤a21-p2-e-a2Tb21-p2-e-a2T+b2p2e-a2T+ε2≙η2+ε2.Through observation of the third equation of (3), we have(17)y2′t≤e-d1τλ1β1f1η1+ε1+λ2β2f2η2+ε2y2t-τ-d2y2t-ry22t.Consider the following differential comparison system:(18)u2′t=e-d1τλ1β1f1η1+ε1+λ2β2f2η2+ε2u2t-τ-d2u2t-ru22t.According to (11) and Lemma 1, we have limt→∞⁡u2(t)=0. Since y2(ξ)=φ3(ξ), ξ∈[-τ,0], u2(t) is the solution of (18) with initial conditions u2(ξ)=φ3(ξ), ξ∈[-τ,0]; by comparison theorem, we have limt→∞⁡y2(t)≤limt→∞⁡u2(t)=0. In view of the positivity of y2(t), we have limt→∞⁡y2(t)=0. It implies that for arbitrarily small positive constant ε3 and t large enough, we have(19)0<y2t<ε3.Further, from the first and the fourth equation of (3), we have(20)x1′t≥x1ta1-β1L1ε3-b1x1t,t≠nT,x1t+=1-p1x1t,t=nT.Considering the following comparison system of (20),(21)u3′t=u3ta1-β1L1ε3-b1u3t,t≠nT,u3t+=1-p1u3t,t=nT,by Lemma 2, we get the positive periodic solution of system (21) as follows:(22)u3∗t=a1-β1L1ε31-p1-e-a1-β1L1ε3Tb11-p1-e-a1-β1L1ε3T+b1p1e-a1-β1L1ε3t-nT,t∈nT,n+1T,with(23)u3∗0+=a1-β1L1ε31-p1-e-a1-β1L1ε3Tb11-e-a1-β1L1ε3T=x1∗0+.By comparison theorem, for given constant ε1>0 and t large enough, we have u3∗(t)-ε1<x1(t). Let ε3→0, then u3∗(t)→x1∗(t), so we have x1∗(t)-ε3<x1(t). It follows from (15) that x1(t)<x1∗(t)+ε1 for t sufficiently large, which implies that x1(t)→x1∗(t) as t→∞. Similarly, we can obtain x2(t)→x2∗(t) as t→∞. This is the end of the proof.


## 4. Permanence of System (3)

Now we investigate the permanence of system (3). Before stating the theorem, we give the definition of permanence for system (3).Definition 4.
System (3) is said to be permanent, if there exist two positive constants m and M, such that, for any solution (x1(t),x2(t),y2(t)) of (3), m≤x1(t), x2(t), y2(t)≤M holds for t sufficiently large.Theorem 5.
Suppose that conditions of (H1) and (H2) hold; moreover if the following conditions:(H4):
1-pi-exp(-(ai-βiLiη3)T)>0, i=1,2,(H5):
e-d1τ(λ1β1f1(ξ1)+λ2β2f2(ξ2))-d2-rη3>0,are satisfied, where η3, ξ1, and ξ2 are defined in (27), (40), and (42), respectively, then system (3) is permanent.Proof.
Firstly, in view of (15) and (16), noticing that positive constants ε1 and ε2 are arbitrarily chosen and can be sufficiently small, we have(24)x1t≤η1,x2t≤η2.Secondly, from the third equation of system (3), we have the following inequality:(25)y2′t≤e-d1τλ1β1f1η1+λ2β2f2η2y2t-τ-ry22t.Considering the following comparison equation,(26)u4′t=e-d1τλ1β1f1η1+λ2β2f2η2u4t-τ-ru42t,by (H4) and Lemma 1, we have limt→∞⁡u4(t)=e-d1τ(λ1β1f1(η1)+λ2β2f2(η2))/r. According to comparison theorem of differential equation, we get(27)y2t≤limt→∞u4t≤e-d1τλ1β1f1η1+λ2β2f2η2r≙η3.Let M=max{η1,η2,η3}, by (24) and (27); then we have x1(t), x2(t), y2(t)≤M.
The following work is to find a constantm>0 with m<M, such that m≤x1(t), m≤x2(t), and  m≤y2(t).
On one hand, from the first and the fourth equation of (3), combining inequality (27), we have(28)x1′t≥x1ta1-β1L1η3-b1x1t,t≠nT,x1t+=1-p1x1t,t=nT.Consider the following comparison system:(29)u5′t=u5ta1-β1L1η3-b1u5t,t≠nT,u5t+=1-p1u5t,t=nT.According to Lemma 2 and (H5), by using comparison theorem, there exists an arbitrarily small constant ε4>0, such that x1(t)≥u5∗(t)-ε4 for t large enough, where u5∗(t) is the unique and globally stable positive periodic solution of (29) with the following form:(30)u5∗t=a1-β1L1η31-p1-exp⁡-a1-β1L1η3Tb11-p1-exp⁡-a1-β1L1η3T+b1p1exp⁡-a1-β1L1η3t-nT,for t∈(nT,(n+1)T], and(31)u5∗0+=a1-β1L1η31-p1-exp⁡-a1-β1L1η3Tb11-exp⁡-a1-β1L1η3T=x1∗0+.By using comparison theorem of impulsive differential equation, we can derive from (30) that(32)x1t≥u5∗t-ε4≥a1-β1L1η31-p1-exp⁡-a1-β1L1η3Tb11-exp⁡-a1-β1L1η3T-ε4≙m1>0for t∈(nT,(n+1)T]. Similarly, we have(33)x2t≥a2-β2L2η31-p2-exp⁡-a2-β2L2η3Tb21-exp⁡-a2-β2L2η3T-ε4≙m2>0.
On the other hand, in order to prove the stability ofy2(t), we define a Lyapunov function as follows:(34)Vt=y2t+exp⁡-d1τ∫t-τtλ1β1f1s+λ2β2f2sy2sds.Calculating the derivative of V(t) along solution y2(t) of system (3), we get(35)V′t=exp⁡-d1τλ1β1f1x1t+λ2β2f2x2t-d2-ry2ty2t.According to (H4), we can choose a positive constant ε5 small enough such that(36)e-d1τλ1β1f1ξ1-ε5+λ2β2f2ξ2-ε5-d2-rη3>0.For some constant y2∗  (0<y2∗<η3), we claim that y2(t)<y2∗ cannot be true for all t>t0. Suppose that the claim is invalid, then there exists a positive constant t0 such that y2(t)<y2∗ for all t>t0. From system (3), we have(37)x1′t≥x1ta1-β1L1y2∗-b1x1t,t≠nT,x1t+=1-p1x1t,t=nT.From the unique solution u6∗(t) of the comparison system of (37), we have x(t)≥u6∗(t)-ε5, for t large enough, where(38)u6∗t=a1-β1L1y2∗1-p1-exp-a1-β1L1y2∗Tb11-p1-exp-a1-β1L1y2∗T+b1p1exp⁡-a1-β1L1y2∗t-nTis the unique solution of the following system:(39)
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,for t∈(nT,(n+1)T], with(40)u6∗0+=a1-β1L1y2∗1-p1-exp-a1-β1L1y2∗Tb11-exp-a1-β1L1y2∗T≙ξ1.Obviously u6∗(t)≥ξ1, t∈(nT,(n+1)T]. Thus the inequality(41)x1t≥u6∗t-ε5≥ξ1-ε5holds for t sufficiently large. Similarly we have(42)x2t≥a2-β2L2y2∗1-p2-exp⁡-a2-β2L2y2∗Tb21-exp⁡-a2-β2L2y2∗T-ε5≙ξ2-ε5.In view of (35), combining (41) and (42), we get(43)V′t≥exp-d1τλ1β1f1ξ1-ε5+λ2β2f2ξ2-ε5-d2-rη3y2t.
Lety2m≙min{y2(t):t1≤t≤t1+τ}. We can prove y2(t)≥y2m for t>t1. Otherwise, there exists t2>0 such that y2(t)≥y2m for t1≤t≤t1+τ+t2, y2(t1+τ+t2)=y2m, and y2′(t1+τ+t2)≤0. However, from (43), we have(44)V′t1+τ+t2≥exp-d1τλ1β1f1ξ1-ε5+λ2β2f2ξ2-ε5-d2-rη3y2m>0.This is a contradiction. Hence, for all t>t1, we have y2(t)≥y2m>0.
In view of (36) and (43), we have V′(t)>0, which leads to V(t)→∞ as t→∞. It is a contradiction with V(t)≤η3+exp(-d1τ)(λ1β1f1(η1)+λ2β2f2(η2))η3τ. This implies that for any positive constant t0, y2(t)<y2∗ cannot be true for all t>t0; then there are the following two cases.
Case (a). y2(t)>y2∗ is true for all t large enough; then our aim is obtained.
Case (b). y2(t) is oscillatory about y2∗; then we define(45)m3=min⁡y2∗2,y2∗exp⁡-d2+ry2∗τ.Now we show y2(t)≥m3. It is clear that there exist two positive constants t~ and ω such that y2(t~)=y2(t~+ω)=y2∗ and y2(t)<y2∗, t∈(t~,t~+ω), with t~ being sufficiently large such that (37) holds for t∈(t~,t~+ω). By the continuous and bounded properties of y2(t), we know that y2(t) is uniformly continuous. Therefore, there exists a constant t3>0 such that y2(t)>y2∗/2 for all t~≤t≤t~+t3. If ω<t3, then y2(t)>y2∗/2; our aim is obtained. If t3<ω<τ, in view of (3), for t~<t<t~+ω, combining assumption y2(t~)=y2∗ and y2(t)<y2∗(t~<t<t~+ω), we have y2′(t)≥-d2y2(t)-ry22(t)≥(-d2-ry2∗)y2(t). Easily we get y2(t)≥y2∗exp(-(d2+ry2∗)τ)≥m3. If ω>τ, analogously we derive that y2(t)≥m3 for t~≤t≤t~+τ. Since the interval [t~,t~+ω] is chosen at random and the choice of m3 is independent with the positive solution of (3), we conclude that y2(t)≥m3 holds for all t large enough.
Based on the above analysis, lettingm=min{m1,m2,m3}, we have m≤x1(t), m≤x2(t), and m≤y2(t). In addition, x1(t),x2(t),y2(t)≤M holds; hence we conclude that system (3) is permanent. The proof is complete.


## 5. Numerical Simulation

For the generalized functional response of (3), there are many functional responses that meet the condition, such as Holling type I, Holling type II, Holling type III, Crowley-Martin type, Beddington-DeAngelis type, Watt type, and Ivlev type. In this section, we choose two concrete functional responses to illustrate the rationality of our results and try to find more dynamical behaviors of system (3). We choose such function response as Holling type II and Beddington-DeAngelis type as follows:(46)f1x1t=x1tc1+c2x1t,f2x2t=x2tc3+c4x2t+c5y2t.Firstly, leta1=0.65, a2=1, b1=0.65, b2=1, d1=0.5, d2=0.7, β1=1, β2=1, λ1=1, λ2=1, p1=0.2, p2=0.2, c1=1, c2=0.8, c3=1, c4=1, c5=1, r=0.25, τ=1, and T=1. By calculation, all parameters satisfy conditions of Theorem 3; then we obtain from Theorem 3 that a predator-extinction solution of system (3) exists, which is globally attractive. By numerical analysis with MATLAB, we get the following simulation figures of a predator-extinction solution and its global attractivity. Figure 1 shows the existence of a predator-extinction solution with only one initial value and Figure 2 shows the attractivity of the predator-extinction solution; that is, regardless of different initial values, species x1, x2, and y2 converge to the predator-extinction solution.Figure 1
Dynamical behaviors of the predator-extinction solution of system (3) with initial value x10=0.1, x2(0)=0.8, y2(0)=0.5 and a1=0.65, a2=1, b1=0.65, b2=1, d1=0.5, d2=0.7, β1=1, β2=1, λ1=1, λ2=1, p1=0.2, p2=0.2, c1=1, c2=0.8, c3=1, c4=1, c5=1, r=0.25, τ=1, and T=1.Figure 2
Dynamical behavior of system (3) with different initial values. These initial values are chosen randomly, and other parameters are the same as those in Figure 1. One can find that the solutions are globally attractive. The difference between Figures 1 and 2 is that more initial values are chosen in Figure 2 to show that the solutions are globally attractive.Secondly, we choose another set of parameters to illustrate the permanence of system (3). Take a1=0.65, a2=1, b1=0.65, b2=1, d1=0.5, d2=0.2, β1=1, β2=1, λ1=1, λ2=1, p1=0.2, p2=0.2, c1=1, c2=0.8, c3=1, c4=1, c5=1, r=0.25, τ=1, and T=1. One can verify that conditions of Theorem 5 are satisfied; then from Theorem 5, system (3) is permanent. By simulation, the results can be indicated clearly by Figure 3. Figure 3(a) shows the permanence of (3) and Figure 3(b) gives a positive periodic solution of this system.Figure 3
The permanence of system (3) with initial values of x1(0)=0.1, x2(0)=0.8, y2(0)=0.5, and d2=0.2, and other parameters are the same as those in Figure 1. Obviously, all these species can coexist and their densities go into a bounded region. (a) Time series of x1, x2, and y2, which indicate that the solution of (3) goes into a bounded region to be permanent. (b) Phase portrait of system (3), which implies a positive periodic solution.
(a)
(b)Thirdly, in view of (H4), we know that pest population will die out ifp1 and p2 are larger than the corresponding threshold. In order to investigate the influence of p1, p2 and time delay τ, we fix the same parameters in Figure 3 as follows. Consider that a1=0.65, a2=1, b1=0.65, b2=1, d1=0.5, d2=0.2, β1=1, β2=1, λ1=1, λ2=1, c1=1, c2=0.8, c3=1, c4=1, c5=1, r=0.25, and T=1. If p1=0.5, by simulation, pest x1 is driven to extinction (see Figure 4(a)), and if p2=0.65, then, similarly, pest x2 becomes extinct (see Figure 4(b)). If p1=0.5 and p2=0.65 at the same time, then not only both pests are going to extinct but also their predator dies out due to lack of food (see Figure 4(c)), which is contrary to the conservation of biological diversity. From biological point of view, we only need to control these two pests at a rational level by adjusting the value of p1 and p2, respectively. Furthermore, by simulation, if time delay τ between immature predator and mature predator goes up to a threshold (τ=4), the predator will die out (see Figure 4(d)), so we claim that the stage-structure also plays an important role in the permanence of system (3).Figure 4
Dynamical behavior of system (3) with initial values x1(0)=0.7, x2(0)=0.8, and y2(0)=0.5. (a) Time series of x1, x2, and y2 with p1=0.5, p2=0.2, which imply that overcatching of x1 will drive it to die out. (b) Time series of x1, x2, and y2 with p1=0.2, p2=0.65, which imply that overcatching of x2 will also drive it to die out. (c) Time series of x1, x2, and y2 with p1=0.5, p2=0.65. Figures show that the three species die out because of overcapturing of the two preys. (d) Time series of x1, x2, and y2 with τ=4, the remaining parameters are as fixed as those in Figure 3. Comparing Figure 4 with Figure 3, one can find that the delay τ large enough can lead y2 to die out.
(a)
(b)
(c)
(d)Finally, we consider the influence of impulsive periodT. Take parameters in system (3) as a1=0.65, a2=1, b1=0.65, b2=1, d1=0.5, d2=0.1, β1=1, β2=1, λ1=1, λ2=1, p1=0.1, p2=0.1, c1=10, c2=0.8, c3=10, c4=1, c5=1, r=0.25, τ=0, x1(0)=0.7, x2(0)=0.8, and y2(0)=0.5. By simulation, we get the following bifurcation diagrams (see Figure 5). Figure 5 indicates that bifurcation appears if T=118.1, 143.9, 147.4, respectively, and if  T∈[118.1,147.4], more than one periodic solution appears. If a moderate pulse is given (T>147), then the system exhibits chaotic phenomena, including stable solutions, cycles, cascade, and chaos, which means the evolution of this system is unpredictable. In a word, the system analyzed here exhibits many complicated dynamical behaviors.Figure 5
Bifurcation diagrams of system (3) with respect to impulsive period T on [107,157]. (a) Bifurcation diagrams of x1. (b) Bifurcation diagrams of x2. (c) Bifurcation diagrams of y2. Figure 5 indicates that bifurcation appears if T=118.1, 143.9, 147.4, respectively, and if T∈[118.1,147.4], more than one periodic solution appears. If a moderate pulse is given (T>147), then the system shows chaotic phenomenon. The bifurcation diagrams include stable solutions, cycles, cascade, and chaos.
(a)
(b)
(c)


## 6. Discussion

In this paper, considering the complicated effects from the real world, we introduce impulsive spraying pesticides, stage-structure for predator, and generalized functional response into one-predator and two-prey system. Firstly, we investigate the existence and global attractivity of predator-extinction periodic solution under the condition thate-d1τ(λ1β1f1(η1)+λ2β2f2(η2))<d2. Secondly, we obtain the sufficient conditions of the permanence. Finally, by numerical simulation with MATLAB, we further discuss some complicated dynamical behaviors of the system.Our obtained results imply that ifd1 or d2 is larger than a threshold (because of lack of food or catching the pest that died from insecticide), the predator will be extinct (see Figure 1), and if pesticides are used too much or harvesting is excessive on two pests, three species will all die out (see Figure 4(c)). In order to keep biological balance or biological diversity, some protective measures can be taken to ensure d2 is less than the threshold (such as disease prevention and releasing immature or mature predator); then the system will be permanent (see Figures 1–3). By comparing Figure 3 with Figures 4(a) and 4(b), if we change parameters p1 and p2, respectively, x1 and x2 will die out effectively, but the rest of population will still survive, which can be used to provide a reliable control strategy: if impulsive period T is given, we can adjust p1, p2 to give a protection for the predator. It will not only reduce the economic loss but also protect environment from damage. Finally, impulsive period T affects the dynamical behaviors of the system heavily, which may bring chaotic phenomena, including stable solutions, cycles, cascade, and chaos (see Figure 5).In a word, our obtained results show that all parametersp1, p2, τ, and T bring great effects on the properties of system (3), which can be applied to ecological resource management. The complicated dynamical behaviors imply that the influence from parameters p1, p2, τ, and T is worthy of being studied and we will continue to study the potential dynamical properties in the near future.



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