# An Impulsive Three-Species Model with Square Root Functional Response and Mutual Interference of Predator

Authors: Zhen Wang; Yuanfu Shao; Xianjia Fang; Xiangmin Ma
Journal: Discrete Dynamics in Nature and Society
 (2016)

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## Abstract

An impulsive two-prey and one-predator model with square root functional responses, mutual interference, and integrated pest management is constructed. By using techniques of impulsive perturbations, comparison theorem, and Floquet theory, the existence and global asymptotic stability of prey-eradication periodic solution are investigated. We use some methods and sufficient conditions to prove the permanence of the system which involve multiple Lyapunov functions and differential comparison theorem. Numerical simulations are given to portray the complex behaviors of this system. Finally, we analyze the biological meanings of these results and give some suggestions for feasible control strategies.


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## Body

## 1. Introduction and Model Formulation

In real world, the study on models of three or more species is very popular, such as food-chain and food webs systems, which have extremely rich dynamics [1, 2]. For predator-prey model, in portrayal of the relationship between predator and prey, a crucial element is the classic definition of a predator’s functional response. In the past few decades, many different functional responses have been extensively investigated [3–7]. For example, Liu et al. [5] gave the following Holling type II functional response which describes the relations of one prey and one predator:(1)x1′t=x1ta-bx1t-αx1tx2t1+wx1t,x2′t=x2tkαx1t1+wx1t-c,where x1(t) is the density of prey and x2(t) is the density of predator at time t. a is the intrinsic growth rate of prey. b represents the rate of intraspecific competition or density dependence. c is the death rate of predator. k is transformation rate for the predator to prey; αx1(t)x2(t)/1+wx1(t) denotes the Holling type II functional response.However, in the actual ecosystem, if examining more complicated ecological case, some preys show herd behavior. That is to say, the predator interacts with the prey along the outer corridor of the herd of prey. Hence actual dynamic behaviors of individuals have not been described in detail by the predation term of Holling type II functional response. Ajraldi et al. [8] pointed out that, by using the terms of the square root of the prey population, the response functions of prey that exhibited herd behavior are more properly modeled. In this respect, Braza [9] gave the following predator-prey model: (2)x1′t=x1ta-bx1t-αx1tx2t1+wx1t,x2′t=x2tkαx1t1+wx1t-c,where αx1(t)/1+wx1(t) is the square root functional response. Ma et al. [10] also investigated a predator-prey system with square root functional response. Their results showed that square root functional response brought about large influence to the dynamical behaviors.On the other hand, few researchers consider the mutual interference between predators, but mutual interference between predators always exists in the actual ecosystem. In 1971, Hassell set about studying the capturing behavior between hosts and parasites; he discovered that hosts or parasites had the tendency to depart from each other when they met, which affected the hosts capturing. If the size of parasite became larger and larger, then the mutual interference would be stronger and stronger. Hence he introduced the mutual interference of predator [11]. Considering the effect from mutual interference between predators, the dynamic behaviors were more complex. For example, He et al. [12] and Zhang et al. [13] investigated the mutual interference of the predator in detail and obtained much different dynamics with those models without mutual interference. Hence, for predator-prey system, it is necessary to consider the mutual interference of predator.Based on above discussion, we give the following prey-predator system with square root functional response and mutual interference of the predator:(3)x1′t=x1ta-bx1t-αx1tx2tm1+wx1t,x2′t=kαx1tx2tm1+wx1t-cx2t,where 0<m≤1 (see [8] for the details).As is known to all, insects have a profound impact on the survival and development of human beings. Most insects are beneficial to human beings; only a few insects are harmful to human life and agricultural development when they reach a certain amount. Hence it is necessary to kill the harmful pests or control them in a certain quantity. Chemical control and biological control are two most commonly used methods. Chemical control is often applied by spraying pesticides, which are used widely because they can kill pests quickly and reduce economic losses in a short time, but they also produce serious environmental pollution. For less pollution to the environment, by stocking or releasing natural enemies, biological control appears, but the effects are not very great. In order to combine different approaches to control pests at the same time, integrated pest management is given to maximize control efficiency and reduce pollution. During the last two decades, ecological pest control is a complex project [14, 15]. For predator-prey system, pest control strategy has been an important topic for many researchers [16, 17].The main purpose of this paper is to investigate the dynamical behaviors of an impulsive one-predator two-prey model with mutual interference, square root functional response, and integrated control methods. The model is described by the following differential equations:(4)x1′t=x1ta1-b1x1t-αx2t-c1x3tmx1t1+d1x1t,x2′t=x2ta2-b2x2t-βx1t-c2x3tmx2t1+d2x2t,x3′t=-a3x3t+c1k1x1t1+d1x1t+c2k2x2t1+d2x2tx3tm,t≠nT,t≠n+l-1T,x1t+=1-μ1x1t,x2t+=1-μ2x2t,x3t+=1-μ3x3t,t=n+l-1T,x1t+=x1t,x2t+=x2t,x3t+=x3t+p,t=nT,where x1(t), x2(t), and x3(t) are densities of two preys and one predator at time t, respectively. ai(i=1,2) is intrinsic increasing rate; a3 is the death rate of predator. m represents the mutual interference of the predator: 0<m≤1. ki(i=1,2) is transformation rate for the predator to prey. bi(i=1,2) is death rate of prey; 0<μi<1(i=1,2,3) represents the percent of prey-predator that dies at time t=(n+l-1)T,0<l<1.p>0 is the releasing number of predators at t=nT. Parameters α,β are competitive effects between two preys, respectively. Parameter T is the moment period of impulsive effect. The integer n∈N; N is the set of all nonnegative integers. All parameters are positive constants.We aim to investigate the dynamical behaviors of (4). From the biological point of view, we only consider system (4) in the biological meaningful region: (5)D=x1t,x2t,x3t:x1t≥0,x2t≥0,x3t≥0,and the initial conditions for system (4) are(6)φ1t,φ2t,φ3t∈C-T,0,R+3,φit>0,R+3=x1,x2,x3∣xi∈R+=0,+∞,i=1,2,3.The structure of this paper is as follows. In Section2, we give some definitions, notations, and lemmas. In Section 3, by using techniques of impulsive perturbations, Floquet theory, and comparison theorem, we discuss stability, extinction, and permanence of system (4). We give corollaries for single chemical control in Section 4. Then we give some examples and numerical analysis of system (4) in Section 5. Finally, we conclude this paper with a brief discussion in Section 6.


## 2. Preliminaries

In this section, some helpful remarks, notations, definitions, and lemmas are introduced which are useful for our main results.Letf=(f1,f2,f3) be the map defined by the right-hand sides of (4). Solution of (4), denoted by X(t)=(x1(t),x2(t),x3(t)):R+→R+3, is continuously differentiable on ((n-1)T,(n+l-1)T) and ((n+l-1)T,nT),(n∈N). Let V:R+×R+3→R+, and then V is said to belong to class V0 if(i)
V is continuous on ((n-1)T,(n+l-1)T]×R+3 and ((n+l-1)T,nT]×R+3,n=1,2,…, and for each X(t)∈R+3, lim(t,y)→((n+l-1)T+,x)V(t,y)=V((n+l-1)T+,x) and lim(t,y)→(nT+,x)V(t,y)=V(nT+,x) exist;(ii)
V is locally Lipschitzian in X(t).Definition 1.
LetV∈V0. For (t,x)∈((n-1)T,(n+l-1)T]×R+3 and ((n+l-1)T,nT]×R+3, the upper right derivative of V with respect to the impulsive differential system (4) is defined as(7)D+Vt,x=limsuph→0+⁡1hVt+h,x+hft,x-Vt,x.Definition 2.
If there exist positive constantsm0 and M0, with each positive solution (x1(t),x2(t),x3(t)) of (4) satisfying m0≤x1(t),x2(t),x3(t)≤M0 for all t sufficiently large, then system (4) is said to be permanent.Remark 3.
The solutionX(t) of (4) is a piecewise continuous function; X(t):R+→R+3, X(t) is continuous on ((n-1)T,(n+l-1)T) and ((n+l-1)T,nT), and X(t0+)=limt→t0+X(t) exists, where t0=(n+l-1)T+ and nT+. The smoothness properties of f ensure the global existence and peculiarity of solutions of (4) (see [18] for the details).Lemma 4 (see [18]).
LetV∈V0, and assume that(8)D+Vt,X≤gt,Vt,X,t≠n+l-1T,t≠nT,Vt,Xt+≤ψn1Vt,X,t=n+l-1T,Vt,Xt+≤ψn2Vt,X,t=nT,where g:R+×R+→R+ and g is continuous on ((n-1)T,(n+l-1)T]×R+ and ((n+l-1)T,nT]×R+ for u∈R+2,n∈N,lim(t,y)→(nT+,u)g(t,y)=g(nT+,u) exists, ψn1,ψn2:R+→R+ are nondecreasing for all n∈N. Let r(t) be the maximal solution of the scalar impulsive differential equation,(9)u′t=gt,ut,t≠n+l-1T,t≠nT,ut+=ψn1ut,t=n+l-1T,ut+=ψn2ut,t=nT,u0+=u0≥0,existing on [0,∞). Then V(0+,X0)≤u0 implies that V(t,X(t))≤r(t) for all t≥0, where X(t) is any solution of system (4).Next, we introduce some fundamental properties about the following subsystem of (4):(10)x3′t=-a3x3t,t≠nT,t≠n+l-1T,x3t+=1-μ3x3t,t=n+l-1T,x3t+=x3t+p,t=nT,x3t=x30.System (10) is a periodically forced linear system easily used to obtain(11)
x
3
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,(n∈N,x3∗0+=x3∗nT+=p/1-1-μ3exp⁡-a3T, x3∗n+l-1T+=x3∗lT+=p1-μ3exp⁡-a3T/1-1-μ3exp⁡-a3T) is a positive periodic solution of system (10). Since the solution of (10) with initial value x30≥0 is(12)
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.Lemma 5 (see [18]).
Supposex3∗(t) is a positive periodic solution of (10) and x30>0; then we get x3t-x3∗(t)→0 as t→+∞.Lemma 6 (see [19]).
SupposeX(t) is a solution of (4) and X(0+)≥0, and hence X(t)≥0 for all t≥0. It also has X(t)>0(t≥0) if X(0+)>0.Lemma 7 (see [20]).
Suppose functionu(t)∈PC1R+,R satisfies the following inequalities:(13)dudt≤ftut+ht,t≠τk,t>0,uτk≤αkuτk+βk,k≥0,u0+≤u0,where f,h∈C(R+,R) and αk≥0, βk, and u0 are constants (k=1,2,…). Then for t>0, one has(14)ut≤u0∏0<τk<tαkexp⁡∫0tfsds+∫0t∏0≤τk<tαkexp⁡∫0tfγdγhdds+∑0<τk<t∏τk<τj<tαjexp⁡∫τktfγdγβk.


## 3. Main Theorems

### 3.1. Boundedness

Theorem 8.
For any solution(x1(t),x2(t),x3(t)) of system (4), there exists a constant M>0, such that x1(t)≤M, x2(t)≤M, and x3(t)≤M hold for all t large enough.Proof.
LetX(t)=(x1(t),x2(t),x3(t)) be a solution of (4) with initial value (x10,x20,x30).
Define a functionω(t)=k1x1(t)+k2x2(t)+x3(t),ω(t)∈V0.
Whent≠(k+l-1)T,kT, take a constant L such that 0<L<a3; then by calculating the upper right derivative of ω(t,X(t)) along the solution of system (4), we have(15)D+ωt+Lωt=k1a1+Lx1t-k1b1x12t-k1αx1tx2t+k2a2+Lx2t-k2b2x22t-k2βx1tx2t+L-a3x3t≤k1a1+Lx1t-k1b1x12t+k2a2+Lx2t-k2b2x22t≤M0,where M0=k12(a1+L)2/4k1b1+k22(a2+L)2/4k2b2.
Further, at momentt=(k+l-1)T,ω((k+l-1)T+)≤ω((k+l-1)T) and at t=kT,ω(kT+)≤ω(kT)+p. Then by Lemma 7, for all t≥0, we get(16)ωt≤ω0e-Lt+k0L1-e-Lt+pe-Lt-T1-eLT+peLTeLT-1⟶k0L+peLTeLT-1ast⟶∞=M1.Hence ω(t) is bounded for sufficiently large t. Let M=minM0,M1⁡; then x1(t), x2t, and x3(t) are bounded by M for sufficiently large t. This completes the proof.


### 3.2. Stability of Prey-Eradication Periodic Solution

Theorem 9.
Suppose(x1(t),x2(t),x3(t)) is any solution of (4); then the prey-eradication periodic solution (0,0,x3∗(t)) is globally asymptotically stable provided that(17)T<min⁡1a1ln⁡11-μ1,1a2ln⁡11-μ2.Proof.
The local stability of periodic solution(0,0,x3∗(t)) can be determined by considering the behavior of small amplitude perturbations of the solution.
Definex1t=u1t, x2t=u2t, and x3(t)=x3∗(t)+u3(t), and we get(18)u1tu2tu3t=Φtu10u20u30,0≤t<T,where ui(t)(i=1,2,3) is a small perturbation. When t≠nT, and t≠(n+l-1)T, (4) can be expanded in a Taylor series. Then, neglecting higher-order terms, the linearized equations read(19)u1′t=a1u1t,u2′t=a2u2t,u3′t=-a3u3t.Let Φ(t) be the fundamental matrix of above differential equations; then Φ(t) satisfies(20)dΦtdt=a1000a2000-a3Φt,Φt=ea1t000ea2t000e-a3t;Φ(0)=I is the identity matrix. Then the linearization of system (4) becomes(21)u1n+l-1T+u2n+l-1T+u3n+l-1T+=1-μ10001-μ20001-μ3u1n+l-1Tu2n+l-1Tu3n+l-1T,u1nT+u2nT+u3nT+=100010001u1nTu2nTu3nT.The stability of periodic solution (0,0,x3∗(t)) of (4) is determined by the eigenvalues of the matrix A:(22)A=1-μ10001-μ20001-μ3100010001ΦT.If each of these eigenvalues of matrix A is less than one, then the periodic solution (0,0,x3∗(t)) is locally stable. The three eigenvalues of the matrix A are(23)λ1=1-μ1exp⁡∫0Ta1dt,λ2=1-μ2exp⁡∫0Ta2dt,λ3=1-μ3exp⁡-a3T.From Floquet theory of impulsive differential equation [18], if λi<1(i=1,2,3), then (0,0,x3∗(t)) is locally asymptotically stable. Here λ3 is already less than one, so we only need to calculate λi<1(i=1,2).
Actually,(24)1-μiexp⁡∫0Taidt<1,i=1,2,⟺∫0Taidt<ln⁡11-μi,i=1,2,⟺aiT<ln⁡11-μi,i=1,2,⟺T<1ailn⁡11-μi,i=1,2.Hence, if T<min1/a1ln⁡1/1-μ1,1/a2ln⁡1/1-μ2, then (0,0,x3∗(t)) is locally asymptotically stable.
Next, we prove the global attractivity of(0,0,x3∗(t)). Choose ε>0 such that(25)ξ1≜1-μ1exp⁡∫0Ta1-b1x1t-εdt<1,ξ2≜1-μ2exp⁡∫0Ta2-b2x2t-εdt<1.From (4), we get (26)dx1tdt≤a1-b1x1tx1t,t≠n+l-1T,x1t+=1-μ1x1t,t=n+l-1T.It follows from (26) that(27)x1n+lT≤x1n+l-1T+exp⁡∫0Ta1-b1x1t-εdt=x1n+l-1T1-μ1exp⁡∫0Ta1-b1x1t-εdt=x1n+l-1Tξ1.Continuing the iteration technique, we can obtain x1((n+l)T)≤x1(lT)ξ1n and x1(lT)ξ1n→0 as n→∞. Hence, x1((n+l)T)→0 as n→∞. Then, x1t→0 as n→∞ since 0<x1t<x1n+l-1T1-μ1expa1T for (n+l-1)T<t≤(n+l)T. By the same way, we can get x2(t)→0, as n→∞.
Now we provex3(t)→x3∗(t) as t→∞. For ε>0(c1k1ε/1+d1ε+c2k2ε/1+d2ε<a3) sufficiently small, there exists T′>0 such that 0<x1(t)<ε and 0<x2(t)<ε,t>T′. Without loss of generality, we assume that 0<x1(t)<ε and 0<x2(t)<ε for all t≥0. Then from system (4) we obtain(28)-a3x3t≤dx3tdt≤-a3x3t+c1k1ε1+d1ε+c2k2ε1+d2εx3tm.Since 0<m≤1, from (28), we have (29)-a3x3t≤dx3tdt≤-a3+c1k1ε1+d1ε+c2k2ε1+d2εx3t.Consider the following two comparison systems:(30)dv1tdt=-a3v1t,t≠nT,t≠n+l-1T,v1t+=1-μ3v1t,t=n+l-1,v1t+=v1t+p,t=nT,v10+=x30+,(31)dv2tdt=-a3+c1k1ε1+d1ε+c2k2ε1+d2εv2t,t≠nT,t≠n+l-1T,v2t+=1-μ3v2t,t=n+l-1,v2t+=v2t+p,t=nT,v20+=x30+.We can obtain that the periodic solution v1∗(t)=x3∗(t), and periodic solution of (31) is (32)
v
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t
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.For any solutions v1(t) and v2(t) of the above two systems, respectively, we have v1(t)→v1∗(t)=x3∗(t), and v2(t)→v2∗(t)→x3∗(t) as ε→0 and t→∞. From Lemmas 5 and 4, we have v1(t)≤x3(t)≤v2(t). Then, for any ε1>0, there exists constant T1>0 such that v1∗(t)-ε1<x3(t)<v2∗(t)+ε1,t≥T1. Let ε→0, and we have x3∗(t)-ε1<x3(t)<x3∗(t)+ε1 for t large enough, which implies x3(t)→x3∗(t) as t→∞. This completes the proof.


### 3.3. Permanence of System (4)

Theorem 10.
System (4) is permanent if conditions (H1) and (H2) hold:(H1)
pm<minB1/c1A1(ln1-μ1+(a1-αa2/b2)T),B2/c2A2(ln1-μ2+(a2-βa1/b1)T),(H2)
a3(b2+d2a2)>c2k2a2,a3(b1+d1a1)>c1k1a1, where(33)A1=1-exp⁡-a3-c2k2a2b2+d2a2mlT+1-μ3mexp⁡-a3-c2k2a2b2+d1a2mlT-exp⁡-a3-c2k2a2b2+d1a2mT,A2=1-exp⁡-a3-c1k1a1b1+d1a1mlT+1-μ3mexp⁡-a3-c1k1a1b1+d1a1mlT-exp⁡-a3-c1k1a1b1+d1a1mT,B1=d1a3-c2k2a2b2+d2a2m1-1-μ3exp⁡-a3-c2k2a2b2+d2a2Tm,B2=d1a3-c1k1a1b1+d1a1m1-1-μ3exp⁡-a3-c1k1a1b1+d1a1Tm.Proof.
SupposeX(t)=(x1(t),x2(t),x3(t)) is any solution of system (4) with X(0)>0. From Theorem 8, we know that there exists constant M>0 such that x1(t)<M,x2(t)<M,x3(t)<M with t≥0. Noticing that dx3(t)/dt≥-a3x3(t), we consider this impulsive differential equation:(34)v3′t=-a3v3t,t≠nT,t≠n+l-1T,v3t+=1-μ3v3t,t=n+l-1T,v3t+=v3t+p,t=nT,v30+=x30+>0.Periodic solution of (34) is v3∗=p(1-μ3)exp(-a3T)/1-(1-μ3)exp(-a3T). By Lemmas 4 and 6, we have x3(t)≥v3(t)>x3∗(t)-ε and x3(t)≥pexp⁡(-r3(t-nT))/1-exp(-r3T)-ε=m3(ε>0) for t sufficiently large. Now we want to find m1~>0,m2~>0 such that x1(t)≥m1~>0,x2≥m2~>0 for t large enough. We prove it by the following two steps.
Step  1. We conclude that there exist m1>0,m2>0,t1,t2∈(0,+∞) such that x1(t1)≥m1,x2(t2)≥m2. Otherwise there will be another three cases.
Case  1. There exists a constant t2>0 such that x2(t2)≥m2, but x1(t)<m1 for all t>0.
Case  2. There exists a constant t1>0 such that x1(t1)≥m1, but x2(t)<m2 for all t>0.
Case  3. Consider x1(t)<m1, and x2(t)<m2 for all t>0.
For Case  1, choose a sufficiently small constantη1>0 such that (35)δ1=1-μ1exp⁡∫n+l-1Tn+lTa1-b1M-αM-c1Md1γ1∗t+η1mdt>1.Then we have (36)x3′=x3k1c1x1x3m-11+d1x1t+k2c2x2x3m-11+d2x2t-a3≤x3k1c1m3m-1M+k2c2m3m-1M-a3≜F1x3t,t≠nT,t≠n+l-1T,x3t+=1-μ3x3t,t=n+l-1T,x3t+=x3t+p,t=nT.Consider the following comparison equation:(37)γ1′=F1γ1,t≠nT,t≠n+l-1T,γ1t+=1-μ3γ1,t=n+l-1T,γ1t+=γ+p,t=nT,γ10+=x30+>0;the periodic solution of (37) reads(38)
γ
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1
-
μ
3
exp
⁡
F
1
T
,
n
-
1
T
<
t
≤
n
+
l
-
1
T
,
p
1
-
μ
3
exp
⁡
F
1
t
-
n
-
1
T
1
-
1
-
μ
3
exp
⁡
F
1
T
,
n
+
l
-
1
T
<
t
≤
n
T
.For any solution γ1(t) of (37), we have γ1(t)→γ1∗(t). By comparison theorem, we can easily obtain x3(t)≤γ1(t) as t→+∞. Hence, there exists T1>0, when t>T1,x3(t)≤γ1(t)<γ1∗(t)+η1.
Then we have(39)x1′t≥x1ta1-b1M-αM-c1Md1γ1∗t+η1m,t≠nT,t≠n+l-1T,x1t+=1-μ1x1t,t=n+l-1T,x1t+=0,t=nT;N1∈N and (N1+l-1)T≥T1. Integrating (39) on ((n+l-1)T,(n+l)T],n≥N1, we get(40)x1n+lT≥x1n+l-1T+exp⁡∫n+l-1Tn+lTa1-b1M-αM-c1d1Mγ1∗t+η1mdt=x1n+l-1T1-μ1exp⁡∫n+l-1Tn+lTa1-b1M-αM-c1d1Mγ1∗t+η1mdt=x1n+l-1Tδ1.Continuing the iteration technique, we can obtain x1((N1+k)T)≥x1((N1)T)σ1k→∞ as k→∞, which is a contradiction. Similar to Case  1, we can prove Case  2 is contradictory.Next we consider Case  3. Using the assumption in Case  3, it is easy to get(41)x3′=x3k1c1x1x3m-11+d1x1t+k2c2x2x3m-11+d2x2t-a3≤x3k1c1Mm-1m1+k2c2Mm-1m2-a3≜x3tF2,t≠nT,t≠n+l-1T,x3t+=1-μ3x3t,t=n+l-1T,x3t+=x3t+p,t=nT.By the same discussion as Case  1, we can find a contradiction. Hence, from the above three cases, there existm1>0 and m2>0,t1,t2∈(0,+∞) such that x1(t1)≥m1,x2(t2)≥m2.Step 2. If x1(t)≥m1 for all t≥t1, then our aim is achieved. Otherwise, if x1(t)<m1 for some t≥t1, then we only need to consider those solutions which leave the region X(t)∈R+3:x1(t)≥m1 and reenter it again. Let t∗=inft≥t1x1(t)<m1, and then there are two possible cases for t∗.Case  1. If t∗=(n1+l-1)T,n1∈N, then x1(t)≥m1 for t∈[t1,t∗) and (1-μ1)m1≤x1(t∗+)=(1-μ1)x1(t∗)<m1. Choose n2,n3∈N such that(42)n2-1T>ln⁡η1/M+pF1,1-μ1n2δ1n3exp⁡δn2T>1-μ1n2δ1n3δn2+1T>1,where δ=a1-b1M-αM-c1MmM/d1<0. Let T¯=n2T+n3T. We claim there must be t′∈(t∗,t∗+T¯) such that x1t′>m1. Otherwise we have x1t′<m1, considering (37) with γ1(n1T+)=x3(n1T+), and then we get(43)
γ
1
t
=
1
-
μ
3
n
-
n
1
+
l
x
3
n
1
T
+
-
p
1
-
1
-
μ
3
exp
⁡
F
1
T
exp
⁡
F
1
t
-
n
1
T
+
γ
1
∗
t
,
n
-
1
T
<
t
≤
n
+
l
-
1
T
,
1
-
μ
3
n
-
n
1
+
l
-
1
x
3
n
1
T
+
-
p
1
-
1
-
μ
3
exp
⁡
F
1
T
exp
⁡
F
1
t
-
n
1
T
+
γ
1
∗
t
,
n
+
l
-
1
T
<
t
≤
n
T
,and n1+1≤n≤n1+n2+n3+1. Then γ1(t)-γ1∗(t)<(M+p)exp(F1(t-n1T))<η1 and x3(t)≤γ1(t)<γ1∗(t)+η1 for t∗+n2T≤t≤t∗+T¯. These imply that (39) holds for t∗+n2T≤t≤t∗+T¯. By the same discussion of Step  1, we obtain x1(t∗+T¯)≥x1(t∗+n2T)δ1n3.From the first equation of (4), we have(44)x1′t≥x1ta1-b1M-αM-c1MmMd1=δx1t,t≠nT,x1t+=1-μ1x1t,t=nT.Integrating (44) on t∈[t∗,t∗+n2T], then x1(t∗+n2T)≥m1(1-μ1)n2exp(δn2T) and x1(t∗+T¯)≥m1(1-μ1)n2δ1n3exp(δn2T)>m1, which is a contradiction with x1(t)<m1. Hence, x1(t)>m1 holds for t∈(t1,t∗). Let t¯=inft≥t∗x1(t)≥m1, and then x1(t)≤m1,x1(t¯)=m1 and x1(t)≥m1(1-μ1)n2+n3exp((n2+n3)δT)≜m1′ for t∗<t<t¯. So x1(t)≥m1′ holds for t∗<t<t¯. If t>t¯, the same arguments can be expanded since x1(t¯)≥m1.Case  2. If t∗≠(n1+l-1)T,n1∈N, then x1(t)≥m1 for t1≤t<t∗ and x1(t∗)=m1.Supposet∗∈((n1′+l-1)T,(n1′+l)T),n1′∈N, and then there are two possible cases for t∈(t∗,(n1′+l)T).Case  2.1. x1(t)≤m1 holds for all t∈t∗,n1′+lT. Similar to Case  1, we can prove that there must exist t1′∈[(n1′+l)T,(n1′+l)T+T¯] such that x1(t1′)>m1; we omit it here.Lett~=inft≥t∗x1(t)>m1, and then x1(t)≤m1 holds for t∗≤t<t~. Then we have x1(t)≥m1(1-μ1)n2+n3+1exp((n2+n3)δT)≜m1~, and x1(t)≥m1~ for t∈(t∗,t~). For t>t~, the same arguments can be followed since x1(t~)≥m1~.Case  2.2. There exist t∗<t<(n1′+l)T such that x1(t)>m1. Let t⌢=inft≥t∗x1(t)>m1, and then x1(t)≤m1 for t∈(t∗,t⌢) and x1(t⌢)=m1. When t∈(t∗,t⌢), (44) holds. Integrating (44) on (t∗,t⌢), then x1(t)≥x1(t∗)exp(δ(t-t∗))≥m1~exp(δT)>m1. Since x1(t⌢)≥m1~ for t>t∗, the same arguments can be used; then x1(t)≥m1~ for all t≥t1.Thus in both cases, we conclude thatx1t⌢≥m1~ holds for all t≥t1. Similarly, we have x2(t)≥m2~ for all t≥t2. Let m4=maxm1~,m2~,m3, and then we have m4<x1,x2,x3<M. The proof is completed.


## 3.1. Boundedness

Theorem 8.
For any solution(x1(t),x2(t),x3(t)) of system (4), there exists a constant M>0, such that x1(t)≤M, x2(t)≤M, and x3(t)≤M hold for all t large enough.Proof.
LetX(t)=(x1(t),x2(t),x3(t)) be a solution of (4) with initial value (x10,x20,x30).
Define a functionω(t)=k1x1(t)+k2x2(t)+x3(t),ω(t)∈V0.
Whent≠(k+l-1)T,kT, take a constant L such that 0<L<a3; then by calculating the upper right derivative of ω(t,X(t)) along the solution of system (4), we have(15)D+ωt+Lωt=k1a1+Lx1t-k1b1x12t-k1αx1tx2t+k2a2+Lx2t-k2b2x22t-k2βx1tx2t+L-a3x3t≤k1a1+Lx1t-k1b1x12t+k2a2+Lx2t-k2b2x22t≤M0,where M0=k12(a1+L)2/4k1b1+k22(a2+L)2/4k2b2.
Further, at momentt=(k+l-1)T,ω((k+l-1)T+)≤ω((k+l-1)T) and at t=kT,ω(kT+)≤ω(kT)+p. Then by Lemma 7, for all t≥0, we get(16)ωt≤ω0e-Lt+k0L1-e-Lt+pe-Lt-T1-eLT+peLTeLT-1⟶k0L+peLTeLT-1ast⟶∞=M1.Hence ω(t) is bounded for sufficiently large t. Let M=minM0,M1⁡; then x1(t), x2t, and x3(t) are bounded by M for sufficiently large t. This completes the proof.


## 3.2. Stability of Prey-Eradication Periodic Solution

Theorem 9.
Suppose(x1(t),x2(t),x3(t)) is any solution of (4); then the prey-eradication periodic solution (0,0,x3∗(t)) is globally asymptotically stable provided that(17)T<min⁡1a1ln⁡11-μ1,1a2ln⁡11-μ2.Proof.
The local stability of periodic solution(0,0,x3∗(t)) can be determined by considering the behavior of small amplitude perturbations of the solution.
Definex1t=u1t, x2t=u2t, and x3(t)=x3∗(t)+u3(t), and we get(18)u1tu2tu3t=Φtu10u20u30,0≤t<T,where ui(t)(i=1,2,3) is a small perturbation. When t≠nT, and t≠(n+l-1)T, (4) can be expanded in a Taylor series. Then, neglecting higher-order terms, the linearized equations read(19)u1′t=a1u1t,u2′t=a2u2t,u3′t=-a3u3t.Let Φ(t) be the fundamental matrix of above differential equations; then Φ(t) satisfies(20)dΦtdt=a1000a2000-a3Φt,Φt=ea1t000ea2t000e-a3t;Φ(0)=I is the identity matrix. Then the linearization of system (4) becomes(21)u1n+l-1T+u2n+l-1T+u3n+l-1T+=1-μ10001-μ20001-μ3u1n+l-1Tu2n+l-1Tu3n+l-1T,u1nT+u2nT+u3nT+=100010001u1nTu2nTu3nT.The stability of periodic solution (0,0,x3∗(t)) of (4) is determined by the eigenvalues of the matrix A:(22)A=1-μ10001-μ20001-μ3100010001ΦT.If each of these eigenvalues of matrix A is less than one, then the periodic solution (0,0,x3∗(t)) is locally stable. The three eigenvalues of the matrix A are(23)λ1=1-μ1exp⁡∫0Ta1dt,λ2=1-μ2exp⁡∫0Ta2dt,λ3=1-μ3exp⁡-a3T.From Floquet theory of impulsive differential equation [18], if λi<1(i=1,2,3), then (0,0,x3∗(t)) is locally asymptotically stable. Here λ3 is already less than one, so we only need to calculate λi<1(i=1,2).
Actually,(24)1-μiexp⁡∫0Taidt<1,i=1,2,⟺∫0Taidt<ln⁡11-μi,i=1,2,⟺aiT<ln⁡11-μi,i=1,2,⟺T<1ailn⁡11-μi,i=1,2.Hence, if T<min1/a1ln⁡1/1-μ1,1/a2ln⁡1/1-μ2, then (0,0,x3∗(t)) is locally asymptotically stable.
Next, we prove the global attractivity of(0,0,x3∗(t)). Choose ε>0 such that(25)ξ1≜1-μ1exp⁡∫0Ta1-b1x1t-εdt<1,ξ2≜1-μ2exp⁡∫0Ta2-b2x2t-εdt<1.From (4), we get (26)dx1tdt≤a1-b1x1tx1t,t≠n+l-1T,x1t+=1-μ1x1t,t=n+l-1T.It follows from (26) that(27)x1n+lT≤x1n+l-1T+exp⁡∫0Ta1-b1x1t-εdt=x1n+l-1T1-μ1exp⁡∫0Ta1-b1x1t-εdt=x1n+l-1Tξ1.Continuing the iteration technique, we can obtain x1((n+l)T)≤x1(lT)ξ1n and x1(lT)ξ1n→0 as n→∞. Hence, x1((n+l)T)→0 as n→∞. Then, x1t→0 as n→∞ since 0<x1t<x1n+l-1T1-μ1expa1T for (n+l-1)T<t≤(n+l)T. By the same way, we can get x2(t)→0, as n→∞.
Now we provex3(t)→x3∗(t) as t→∞. For ε>0(c1k1ε/1+d1ε+c2k2ε/1+d2ε<a3) sufficiently small, there exists T′>0 such that 0<x1(t)<ε and 0<x2(t)<ε,t>T′. Without loss of generality, we assume that 0<x1(t)<ε and 0<x2(t)<ε for all t≥0. Then from system (4) we obtain(28)-a3x3t≤dx3tdt≤-a3x3t+c1k1ε1+d1ε+c2k2ε1+d2εx3tm.Since 0<m≤1, from (28), we have (29)-a3x3t≤dx3tdt≤-a3+c1k1ε1+d1ε+c2k2ε1+d2εx3t.Consider the following two comparison systems:(30)dv1tdt=-a3v1t,t≠nT,t≠n+l-1T,v1t+=1-μ3v1t,t=n+l-1,v1t+=v1t+p,t=nT,v10+=x30+,(31)dv2tdt=-a3+c1k1ε1+d1ε+c2k2ε1+d2εv2t,t≠nT,t≠n+l-1T,v2t+=1-μ3v2t,t=n+l-1,v2t+=v2t+p,t=nT,v20+=x30+.We can obtain that the periodic solution v1∗(t)=x3∗(t), and periodic solution of (31) is (32)
v
2
∗
t
=
p
exp
⁡
-
a
3
+
c
1
k
1
ε
/
1
+
d
1
ε
+
c
2
k
2
ε
/
1
+
d
2
ε
t
-
n
-
1
T
1
-
1
-
μ
3
exp
⁡
-
a
3
+
c
1
k
1
ε
/
1
+
d
1
ε
+
c
2
k
2
ε
/
1
+
d
2
ε
T
,
n
-
1
T
<
t
≤
n
+
l
-
1
T
,
p
1
-
μ
3
exp
⁡
-
a
3
+
c
1
k
1
ε
/
1
+
d
1
ε
+
c
2
k
2
ε
/
1
+
d
2
ε
t
-
n
-
1
T
1
-
1
-
μ
3
exp
⁡
-
a
3
+
c
1
k
1
ε
/
1
+
d
1
ε
+
c
2
k
2
ε
/
1
+
d
2
ε
T
,
n
+
l
-
1
T
<
t
≤
n
T
.For any solutions v1(t) and v2(t) of the above two systems, respectively, we have v1(t)→v1∗(t)=x3∗(t), and v2(t)→v2∗(t)→x3∗(t) as ε→0 and t→∞. From Lemmas 5 and 4, we have v1(t)≤x3(t)≤v2(t). Then, for any ε1>0, there exists constant T1>0 such that v1∗(t)-ε1<x3(t)<v2∗(t)+ε1,t≥T1. Let ε→0, and we have x3∗(t)-ε1<x3(t)<x3∗(t)+ε1 for t large enough, which implies x3(t)→x3∗(t) as t→∞. This completes the proof.


## 3.3. Permanence of System (4)

Theorem 10.
System (4) is permanent if conditions (H1) and (H2) hold:(H1)
pm<minB1/c1A1(ln1-μ1+(a1-αa2/b2)T),B2/c2A2(ln1-μ2+(a2-βa1/b1)T),(H2)
a3(b2+d2a2)>c2k2a2,a3(b1+d1a1)>c1k1a1, where(33)A1=1-exp⁡-a3-c2k2a2b2+d2a2mlT+1-μ3mexp⁡-a3-c2k2a2b2+d1a2mlT-exp⁡-a3-c2k2a2b2+d1a2mT,A2=1-exp⁡-a3-c1k1a1b1+d1a1mlT+1-μ3mexp⁡-a3-c1k1a1b1+d1a1mlT-exp⁡-a3-c1k1a1b1+d1a1mT,B1=d1a3-c2k2a2b2+d2a2m1-1-μ3exp⁡-a3-c2k2a2b2+d2a2Tm,B2=d1a3-c1k1a1b1+d1a1m1-1-μ3exp⁡-a3-c1k1a1b1+d1a1Tm.Proof.
SupposeX(t)=(x1(t),x2(t),x3(t)) is any solution of system (4) with X(0)>0. From Theorem 8, we know that there exists constant M>0 such that x1(t)<M,x2(t)<M,x3(t)<M with t≥0. Noticing that dx3(t)/dt≥-a3x3(t), we consider this impulsive differential equation:(34)v3′t=-a3v3t,t≠nT,t≠n+l-1T,v3t+=1-μ3v3t,t=n+l-1T,v3t+=v3t+p,t=nT,v30+=x30+>0.Periodic solution of (34) is v3∗=p(1-μ3)exp(-a3T)/1-(1-μ3)exp(-a3T). By Lemmas 4 and 6, we have x3(t)≥v3(t)>x3∗(t)-ε and x3(t)≥pexp⁡(-r3(t-nT))/1-exp(-r3T)-ε=m3(ε>0) for t sufficiently large. Now we want to find m1~>0,m2~>0 such that x1(t)≥m1~>0,x2≥m2~>0 for t large enough. We prove it by the following two steps.
Step  1. We conclude that there exist m1>0,m2>0,t1,t2∈(0,+∞) such that x1(t1)≥m1,x2(t2)≥m2. Otherwise there will be another three cases.
Case  1. There exists a constant t2>0 such that x2(t2)≥m2, but x1(t)<m1 for all t>0.
Case  2. There exists a constant t1>0 such that x1(t1)≥m1, but x2(t)<m2 for all t>0.
Case  3. Consider x1(t)<m1, and x2(t)<m2 for all t>0.
For Case  1, choose a sufficiently small constantη1>0 such that (35)δ1=1-μ1exp⁡∫n+l-1Tn+lTa1-b1M-αM-c1Md1γ1∗t+η1mdt>1.Then we have (36)x3′=x3k1c1x1x3m-11+d1x1t+k2c2x2x3m-11+d2x2t-a3≤x3k1c1m3m-1M+k2c2m3m-1M-a3≜F1x3t,t≠nT,t≠n+l-1T,x3t+=1-μ3x3t,t=n+l-1T,x3t+=x3t+p,t=nT.Consider the following comparison equation:(37)γ1′=F1γ1,t≠nT,t≠n+l-1T,γ1t+=1-μ3γ1,t=n+l-1T,γ1t+=γ+p,t=nT,γ10+=x30+>0;the periodic solution of (37) reads(38)
γ
1
∗
t
=
p
exp
⁡
F
1
t
-
n
-
1
T
1
-
1
-
μ
3
exp
⁡
F
1
T
,
n
-
1
T
<
t
≤
n
+
l
-
1
T
,
p
1
-
μ
3
exp
⁡
F
1
t
-
n
-
1
T
1
-
1
-
μ
3
exp
⁡
F
1
T
,
n
+
l
-
1
T
<
t
≤
n
T
.For any solution γ1(t) of (37), we have γ1(t)→γ1∗(t). By comparison theorem, we can easily obtain x3(t)≤γ1(t) as t→+∞. Hence, there exists T1>0, when t>T1,x3(t)≤γ1(t)<γ1∗(t)+η1.
Then we have(39)x1′t≥x1ta1-b1M-αM-c1Md1γ1∗t+η1m,t≠nT,t≠n+l-1T,x1t+=1-μ1x1t,t=n+l-1T,x1t+=0,t=nT;N1∈N and (N1+l-1)T≥T1. Integrating (39) on ((n+l-1)T,(n+l)T],n≥N1, we get(40)x1n+lT≥x1n+l-1T+exp⁡∫n+l-1Tn+lTa1-b1M-αM-c1d1Mγ1∗t+η1mdt=x1n+l-1T1-μ1exp⁡∫n+l-1Tn+lTa1-b1M-αM-c1d1Mγ1∗t+η1mdt=x1n+l-1Tδ1.Continuing the iteration technique, we can obtain x1((N1+k)T)≥x1((N1)T)σ1k→∞ as k→∞, which is a contradiction. Similar to Case  1, we can prove Case  2 is contradictory.Next we consider Case  3. Using the assumption in Case  3, it is easy to get(41)x3′=x3k1c1x1x3m-11+d1x1t+k2c2x2x3m-11+d2x2t-a3≤x3k1c1Mm-1m1+k2c2Mm-1m2-a3≜x3tF2,t≠nT,t≠n+l-1T,x3t+=1-μ3x3t,t=n+l-1T,x3t+=x3t+p,t=nT.By the same discussion as Case  1, we can find a contradiction. Hence, from the above three cases, there existm1>0 and m2>0,t1,t2∈(0,+∞) such that x1(t1)≥m1,x2(t2)≥m2.Step 2. If x1(t)≥m1 for all t≥t1, then our aim is achieved. Otherwise, if x1(t)<m1 for some t≥t1, then we only need to consider those solutions which leave the region X(t)∈R+3:x1(t)≥m1 and reenter it again. Let t∗=inft≥t1x1(t)<m1, and then there are two possible cases for t∗.Case  1. If t∗=(n1+l-1)T,n1∈N, then x1(t)≥m1 for t∈[t1,t∗) and (1-μ1)m1≤x1(t∗+)=(1-μ1)x1(t∗)<m1. Choose n2,n3∈N such that(42)n2-1T>ln⁡η1/M+pF1,1-μ1n2δ1n3exp⁡δn2T>1-μ1n2δ1n3δn2+1T>1,where δ=a1-b1M-αM-c1MmM/d1<0. Let T¯=n2T+n3T. We claim there must be t′∈(t∗,t∗+T¯) such that x1t′>m1. Otherwise we have x1t′<m1, considering (37) with γ1(n1T+)=x3(n1T+), and then we get(43)
γ
1
t
=
1
-
μ
3
n
-
n
1
+
l
x
3
n
1
T
+
-
p
1
-
1
-
μ
3
exp
⁡
F
1
T
exp
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F
1
t
-
n
1
T
+
γ
1
∗
t
,
n
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1
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<
t
≤
n
+
l
-
1
T
,
1
-
μ
3
n
-
n
1
+
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,and n1+1≤n≤n1+n2+n3+1. Then γ1(t)-γ1∗(t)<(M+p)exp(F1(t-n1T))<η1 and x3(t)≤γ1(t)<γ1∗(t)+η1 for t∗+n2T≤t≤t∗+T¯. These imply that (39) holds for t∗+n2T≤t≤t∗+T¯. By the same discussion of Step  1, we obtain x1(t∗+T¯)≥x1(t∗+n2T)δ1n3.From the first equation of (4), we have(44)x1′t≥x1ta1-b1M-αM-c1MmMd1=δx1t,t≠nT,x1t+=1-μ1x1t,t=nT.Integrating (44) on t∈[t∗,t∗+n2T], then x1(t∗+n2T)≥m1(1-μ1)n2exp(δn2T) and x1(t∗+T¯)≥m1(1-μ1)n2δ1n3exp(δn2T)>m1, which is a contradiction with x1(t)<m1. Hence, x1(t)>m1 holds for t∈(t1,t∗). Let t¯=inft≥t∗x1(t)≥m1, and then x1(t)≤m1,x1(t¯)=m1 and x1(t)≥m1(1-μ1)n2+n3exp((n2+n3)δT)≜m1′ for t∗<t<t¯. So x1(t)≥m1′ holds for t∗<t<t¯. If t>t¯, the same arguments can be expanded since x1(t¯)≥m1.Case  2. If t∗≠(n1+l-1)T,n1∈N, then x1(t)≥m1 for t1≤t<t∗ and x1(t∗)=m1.Supposet∗∈((n1′+l-1)T,(n1′+l)T),n1′∈N, and then there are two possible cases for t∈(t∗,(n1′+l)T).Case  2.1. x1(t)≤m1 holds for all t∈t∗,n1′+lT. Similar to Case  1, we can prove that there must exist t1′∈[(n1′+l)T,(n1′+l)T+T¯] such that x1(t1′)>m1; we omit it here.Lett~=inft≥t∗x1(t)>m1, and then x1(t)≤m1 holds for t∗≤t<t~. Then we have x1(t)≥m1(1-μ1)n2+n3+1exp((n2+n3)δT)≜m1~, and x1(t)≥m1~ for t∈(t∗,t~). For t>t~, the same arguments can be followed since x1(t~)≥m1~.Case  2.2. There exist t∗<t<(n1′+l)T such that x1(t)>m1. Let t⌢=inft≥t∗x1(t)>m1, and then x1(t)≤m1 for t∈(t∗,t⌢) and x1(t⌢)=m1. When t∈(t∗,t⌢), (44) holds. Integrating (44) on (t∗,t⌢), then x1(t)≥x1(t∗)exp(δ(t-t∗))≥m1~exp(δT)>m1. Since x1(t⌢)≥m1~ for t>t∗, the same arguments can be used; then x1(t)≥m1~ for all t≥t1.Thus in both cases, we conclude thatx1t⌢≥m1~ holds for all t≥t1. Similarly, we have x2(t)≥m2~ for all t≥t2. Let m4=maxm1~,m2~,m3, and then we have m4<x1,x2,x3<M. The proof is completed.


## 4. Single Chemical Control

Ifp=0, system (4) concerns the single chemical control. Then we have the following corollary.Corollary 11.
Assumep=0 and l=1.(i)
Let(x1(t),x2(t),x3(t)) be any solution of (4). Then the solution (0,0,x3∗(t)) is locally stable if (45)T<min⁡1a1ln⁡11-μ1,1a2ln⁡11-μ2.(ii)
System (4) is permanent if a3(b2d1+a2d3)>c2k2a2,a3(b1d1+a1d2)>c1k1a1 and(46)T>max⁡1a1-αa2/b2ln⁡11-μ1,1a2-βa1/b1ln⁡11-μ2.


## 5. Examples and Simulations

In this section, we give some examples and numerical simulations to verify our theoretical results and further study the complexity and variety of system (4).Firstly, leta1=0.2, a2=0.3, a3=0.2, b1=0.03, b2=0.04, c1=0.1, c2=0.1, d1=100, d2=100, μ1=0.56, μ2=0.55, μ3=0.09, α=0.003, β=0.004, k1=0.75, k2=0.75, m=1, l=0.2, T=10, and p=1. By verification, these parameters satisfy conditions of Theorem 9; then the prey-eradication periodic solution is global asymptotic stability. With Matlab, by simulation, the asymptotic stability can be illustrated by Figures 1(a), 1(b), and 2. Figures 1(a) and 1(b) show the existence of the prey-eradication periodic solution. Figure 2 shows the attractivity of prey-eradication periodic solution. If we change the mutual interference of the predator m and other parameters are the same as Figure 1(a), then we get another dynamical portrait that is very different from Figure 1(a). We give an example for m=0.6 (see Figure 1(c)), which shows that the mutual interference of the predator m affects the dynamical behaviors of system (4) to be more complex.Figure 1
A prey-eradication periodic solution with initial values ofx1(0)=2, x2(0)=2, and x3(0)=3; (a) time series of x1(t),x2(t),x3(t), which are simulated 100 cycles in the interval [0,200]; (b) phase portrait of x1(t),x2(t),x3(t): the solution goes gradually into the stable state from the initial point [2,2,3]; (c) the dynamic behaviors of system (4) with mutual interference of predator m as m=0.6.
(a)
(b)
(c)Figure 2
Dynamic behavior of the prey-eradication periodic solution with different initial values and the parameters are same as Figure1. (a) Phase portrait of x1(t) and x3(t) with different initial values. (b) Phase portrait of x2(t) and x3(t) with different initial values.
(a)
(b)Secondly, letx1(t)=2, x2(t)=2, x3(t)=3, a1=0.6, a2=0.8, a3=0.2, b1=0.03, b2=0.07, c1=0.8, c2=0.8, d1=30, d2=30, μ1=0.4, μ2=0.4, μ3=0.08, α=0.005, β=0.005, k1=0.8, k2=0.8, m=1, l=1, T=10, and p=10. By verification, these parameters satisfy conditions of Theorem 10. Then we conclude that system (4) is permanent from Theorem 10. By simulation, we can verify the same results. Figure 3(a) is the time series of the three species in the interval [0,200]. In addition, we give the phase portrait of a positive periodic solution of system (4) in Figure 3(b). Similarly, if we change the mutual interference of the predator m and other parameters are same as Figure 3(a), then we get another dynamical picture that is very different from Figure 3(a). We give an example for m=0.66 (see Figure 3(c)). It also shows that the parameter m affects the dynamical behaviors of system (4).Figure 3
(a) is the existence of the positive periodic solution of system (4) with initial values of x1(t)=2,x2(t)=2,x3(t)=3, which implies that system (4) is permanent; (b) is phase portrait of positive periodic solution of system (4); (c) is dynamic behavior of system (4) with the mutual interference of the predator m=0.66; other parameters are also the same as (a).
(a)
(b)
(c)Thirdly, by numerical analysis, we aim to investigate the bifurcation diagrams of impulsive periodT. Let x1(0)=2, x2(0)=2, and x3(0)=3, and a1=0.6, a2=0.8, a3=0.2, b1=0.03, b2=0.07, c1=0.8, c2=0.8, d1=30, d2=30, μ1=0.4, μ2=0.4, μ3=0.08, α=0.005, β=0.005, k1=0.8, k2=0.8, m=1, l=1, and p=10. The bifurcation diagrams of x1(t), x2(t), and x3(t) with respect to parameter T in range [107,155] are shown in Figure 4. Figures 4(a)–4(c) are bifurcation diagrams of x1(t), x2(t), and x3(t), respectively. From these diagrams, we can see that impulsive period T heavily affects the dynamical behaviors. For example, Figure 5 shows the complex dynamic behaviors of x3. Figure 5 is the magnified parts of Figure 4(c). From Figure 5(a), we can see the following: when impulsive period 107<T<117, it is stable; if T=117, 142.4, 148, and 149.1, then bifurcation appears, respectively. That is to say, when T>117, there is a cascade of period-doubling bifurcations leading to 2T-period solution (Figure 5(a)); when T>142.4, 4T-period solution appears (Figure 5(a)); when T>148, 8T-period solution appears (Figure 5(b)); when T>149.1, 16T-period solution appears (Figure 5(b)). As T increase beyond 149.5, the phenomenon of “crisis” emerges. When T=151.6, there is a typical chaotic oscillation (Figure 5(c)). When T is near 153.5, we can see in the neighborhood of T=153.5, after the period-doubling bifurcations, the symmetry-breaking bifurcations appear (Figure 5(d)), which are specially simple bifurcations that come into being multiplicity of steady states [5]. It implies that when 107<T<117, that is, spraying pesticides and releasing natural enemies are frequent, the solution of this system is stable in this situation; when 117<T<149.5, periodic behaviors of prey and the predator will appear; if 149.5<T<155, then dynamical properties of this system are complex and the development of this system may be unpredictable.Figure 4
Bifurcation diagrams of system (4) with x1(0)=0.7,x2(0)=0.8,x3(0)=0.5. The system exhibits stable solution, periodic doubling bifurcation, and chaotic phenomenon corresponding with different T. (a) is the bifurcation diagram of population x1(t); (b) is the bifurcation diagram of population x2(t); (c) is the bifurcation diagram of population x3(t).
(a)
(b)
(c)Figure 5
The magnified parts of Figure4(c) of dynamic behavior of x3. (a) and (b) are period solutions. (c) is a typical chaotic oscillation when T=151.6. (d) is the symmetry-breaking bifurcations when T=153.5.
(a)
(b)
(c)
(d)Fourthly, letx1(t)=0.6, x2(t)=1, and x3(t)=1 and a1=0.6, a2=0.65, b1=0.86, b2=0.83, μ1=0.01, μ2=0.01, α=0.01, and β=0.01. By verification, these parameters satisfy conditions of Corollary 11; then we can get the globally asymptotic stability of prey-eradication periodic solution and permanence for single chemical control (see Figures 6(a) and 6(b)).Figure 6
Dynamical behaviors of prey-eradication periodic solution and permanence of the system with single chemical control: (a) is the prey-eradication periodic solution of the single chemical control; (b) is the permanence of the single chemical control.
(a)
(b)


## 6. Conclusion

In this paper, an impulsive two-prey and one-predator system with square root functional responses, mutual interference, and integrated pest management is constructed. Numerical simulations are given to portray the complex behaviors of this system. From Theorem9, the existence and global asymptotic stability of prey-eradication periodic solution of (4) are obtained. Some methods and sufficient conditions are given to prove the permanence of system (4) in Theorem 10.From Theorems9 and 10 and simulations, we know dynamical properties of system (4) are very complex which depend on impulsive period T, the releasing amount of predator p, the mutual interference of predator m, and the parameter μi(i=1,2,3) of pests or predator which dies from the chemical control. Figures 1(c) and 3(c) show that different values of mutual interference of predator have different dynamical properties for system (4). Figures 1(a), 1(b), and 2 show the existence and attractivity of prey-eradication periodic solution. Figures 3(a) and 3(b) show the existence of positive periodic solution. Figure 4 implies that impulsive period T heavily influences the dynamic behavior of system (4). As T changes, periodic behaviors, bifurcations, “crisis” phenomenon, chaotic phenomenon, chaotic oscillation, and symmetry-breaking bifurcations appear, respectively. Hence, we can choose moderate value of T for some different control strategies.In this paper, by considering mutual interference and square root functional responses, our constructed model is new and complex, which more rationally reflects the real world. Furthermore, we also give the corresponding results on strategies of integrated pest management and classical chemical control. By our obtained results, if combining biological control and chemical control to eradicate preys, we can choose moderate impulsive periodT and moderate parameters p and μi(i=1,2,3) to effectively eliminate preys and reduce environmental pollution. In particular, we analyze the influence from impulsive period T. These theories have some guidance to our real life and the natural balance.



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