AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/324912 324912 Research Article Dynamics of an Almost Periodic Food Chain System with Impulsive Effects Li Yaqin 1 Wu Wenquan 2 Zhang Tianwei 3 Xia Yonghui 1 Department of Mathematics, Kunming University, Kunming 650031 China kmust.edu.cn 2 Department of Mathematics, Aba Teachers College, Wenchuan, Sichuan 623002 China 3 City College, Kunming University of Science and Technology, Kunming 650051 China kmust.edu.cn 2014 20 7 2014 2014 05 06 2014 06 07 2014 21 7 2014 2014 Copyright © 2014 Yaqin Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In order to obtain a more accurate description of the ecological system perturbed by human exploitation activities such as planting and harvesting, we need to consider the impulsive differential equations. Therefore, by applying the comparison theorem and the Lyapunov method of the impulsive differential equations, this paper gives some new sufficient conditions for the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution in a food chain system with almost periodic impulsive perturbations. The method used in this paper provides a possible method to study the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of the models with impulsive perturbations in biological populations. Finally, an example and numerical simulations are given to illustrate that our results are feasible.

1. Introduction

Let R and Z denote the sets of real numbers and integers, respectively. Related to a continuous function f, we use the following notations: (1)fl=infsRf(s),fu=supsRf(s).

As was pointed out by Berryman , the dynamic relationship between predators and their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. Food chain predator-prey system, as one of the most important predator-prey systems, has been extensively studied by many scholars; many excellent results were concerned with the persistent property and positive periodic solution of the system; see  and the references cited therein. Recently, Shen considered the following three species food chain predator-prey system with Holling type IV functional response: (2)x˙1(t)=x1(t)[r1(t)-a1(t)x1(t)-b1(t)x2(t)m1+x12(t)],x˙2(t)=x2(t)[-r2(t)+b2(t)x1(t)m1+x12(t)kkkjjjjjj-a2(t)x2(t)-b3(t)x3(t)m2+x22(t)],x˙3(t)=x3(t)[-r3(t)+b4(t)x2(t)m2+x22(t)-a3(t)x3(t)], where xi(t), i=1,2,3, denotes the density of species Xi at time t, X2 is the predator of the first species X1, and X3 is the predator of the second species X2. By applying the comparison theorem of the differential equation and constructing the suitable Lyapunov function, sufficient conditions which guarantee the permanence and the global attractivity of the system are obtained.

Considering the exploited predator-prey system (harvesting or stocking) is very valuable, for it involves the human activities. It can be referred to , in which the human activities always happen in a short time or instantaneously. The continuous action of human is then removed from the model and replaced with an impulsive perturbation. These models are subject to short-term perturbations which are often assumed to be in the form of impulsive in the modelling process. Consequently, impulsive differential equations provide a natural description of such systems . Then, in , Zhang and Tan studied the following Holling II functional responses food chain system with periodic constant impulsive perturbation of predator: (3)x˙1(t)=x1(t)[1-x1(t)-b1x2(t)1+x1(t)],x˙2(t)=x2(t)[-r2+b2x1(t)1+x1(t)-b3x3(t)1+x2(t)],x˙3(t)=x3(t)[-r3+b4x2(t)1+x2(t)],tnT,Δx1(nT)=0,Δx2(nT)=h,Δx3(nT)=0,n{0,1,}=Z+, where h>0 is the release amount of top predator at t=nT and T is the period of the impulsive effect. By using the Floquet theory of impulsive differential equation and small amplitude perturbation skills, we consider the local stability of prey and top predator eradication periodic solution.

In real world phenomenon, the environment varies due to the factors such as seasonal effects of weather, food supplies, mating habits, and harvesting. So, it is usual to assume the periodicity of parameters in system (2). However, if the various constituent components of the temporally nonuniform environment are with incommensurable (nonintegral multiples) periods, then one has to consider the environment to be almost periodic since the assumption of almost periodicity is more realistic, more important, and more general when we consider the effects of the environmental factors. In recent years, there have been many mathematical studies for the existence, uniqueness, and stability of positive almost periodic solution of biological models governed by differential equations in the literature (see [11, 1525] and the references cited therein). Therefore, Bai and wang in  studied the following nonautonomous food chains system with Holling's type II functional response: (4)x˙1(t)=x1(t)[r1(t)-a1(t)x1(t)-b1(t)x2(t)m1+x1(t)],x˙2(t)=x2(t)[-r2(t)+b2(t)x1(t)m1+x1(t)-a2(t)x2(t)hhhhhh-b3(t)x3(t)m2+x2(t)],x˙3(t)=x3(t)[-r3(t)+b4(t)x2(t)m2+x2(t)-a3(t)x3(t)]. By applying the comparison theorem and the Lyapunov method of ordinary differential equations, some sufficient conditions which guarantee the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of system (4) are obtained.

Stimulated by the above reason, this paper is concerned with the following almost periodic food chain system with almost periodic impulsive perturbations and general functional responses: (5)x˙1(t)=x1(t)[r1(t)-a1(t)x1(t)-b1(t)x2(t)m1+x1α1(t)],x˙2(t)=x2(t)[-r2(t)+b2(t)x1(t)m1+x1α1(t)-a2(t)x2(t)kkkkkkkkkkkkh-b3(t)x3(t)m2+x2α2(t)],x˙3(t)=x3(t)[-r3(t)+b4(t)x2(t)m2+x2α2(t)-a3(t)x3(t)],hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhtτk,Δx1(τk)=h1kx1(τk),Δx2(τk)=h2kx2(τk),Δx3(τk)=h3kx3(τk),kZ+, where ri(t), ai(t), and bj(t), i=1,2,3, j=1,2,3,4, are all continuous almost periodic functions which are bounded above and below by positive constants; m1,m2,α1,α2 are positive constants; h1k,h2k,h3k>-1 are constants; 0=τ0<τ1<τ2<<τk<τk+1< are impulse points with limk+τk=+; and the set of sequences {τkj},τkj=τk+j-τk,kZ+,jZ, is uniformly almost periodic (see Definition 1 in Section 2).

Obviously, system (2)–(4) is special case of system (5).

The main purpose of this paper is to establish some new sufficient conditions which guarantee the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of system (5) by using the comparison theorem and the Lyapunov method of the impulsive differential equations [10, 11] (see Theorems 11 and 14 in Sections 3 and 4).

The organization of this paper is as follows. In Section 2, we give some basic definitions and necessary lemmas which will be used in later sections. In Section 3, by using the comparison theorem of the impulsive differential equations , we give the permanence of system (5). In Section 4, we study the existence of a unique uniformly asymptotically stable positive almost periodic solution of system (5) by applying the Lyapunov method of the impulsive differential equations .

2. Preliminaries

Now, let us state the following definitions and lemmas, which will be useful in proving our main result.

By I, I={{τk}R:τk<τk+1, kZ, limk±τk=±}, we denote the set of all sequences that are unbounded and strictly increasing. Introduce the following notations.

For JR3, PC(J,R3) is the space of all piecewise continuous functions from J to R3 with points of discontinuity of the first kind τk, at which it is left continuous. By the basic theories of impulsive differential equations in [10, 11], system (5) has a unique solution X(t)=X(t,X0)PC([0,+),R3).

Since the solution of system (5) is a piecewise continuous function with points of discontinuity of the first kind τk, kZ, we adopt the following definitions for almost periodicity.

Definition 1 (see [<xref ref-type="bibr" rid="B23">11</xref>]).

The set of sequences {τkj},τkj=τk+j-τk,kZ,jZ,{τk}I, is said to be uniformly almost periodic if for arbitrary ϵ>0 there exists a relatively dense set of ϵ-almost periods common for any sequences.

Definition 2 (see [<xref ref-type="bibr" rid="B23">11</xref>]).

The function φPC(R,R) is said to be almost periodic, if the following hold.

The set of sequences {τkj},τkj=τk+j-τk,kZ,jZ,{τk}I, is uniformly almost periodic.

For any ϵ>0, there exists a real number δ>0 such that if the points t and t′′ belong to one and the same interval of continuity of φ(t) and satisfy the inequality |t-t′′|<δ, then |φ(t)-φ(t′′)|<ϵ.

For any ϵ>0, there exists a relatively dense set T such that if ηT, then |φ(t+η)-φ(t)|<ϵ for all tR satisfying condition |t-τk|>ϵ, kZ. The elements of T are called ϵ-almost periods.

Lemma 3 (see [<xref ref-type="bibr" rid="B23">11</xref>]).

Let {τk}I. Then, there exists a positive integer A such that, on each interval of length 1, one has no more than A elements of the sequence {τk}; that is, (6)i(s,t)A(t-s)+A, where i(s,t) is the number of the points τk in the interval (s,t).

Theoretically, one can investigate the existence, uniqueness, and stability of almost periodic solution for functional differential equations by using Lyapunov functional as follows [11, P109].

Consider the system of impulsive differential equations as follows: (7)x˙(t)=f(t,x(t)),tτk,Δx(τk)=Ikx(τk), where tR, {τk}I, f:R×DRn, Ik:DRn, kZ, and D is an open set in Rn.

Introduce the following conditions.

Function f(t,x) is almost periodic in t uniformly with respect to xD.

Sequence {Ik(x)}, kZ, is almost periodic uniformly with respect to xD.

Lemma 4 (see [<xref ref-type="bibr" rid="B23">11</xref>, P<sub>109</sub>]).

Suppose that there exists a Lyapunov functional V(t,x,y) defined on R+×D×D satisfying the following conditions.

u(x-y)V(t,x,y)v(x-y), where u,vP with P={u:R+R+u is continuous increasing function and u(s)0 as s0}.

|V(t,x-,y-)-V(t,x^,y^)|K(x--x^+y--y^), where K>0 is a constant.

For t=τk, V(t+,x+Ik(x),y+Ik(y))V(t,x,y); for tτk, V˙(2.2)(t,x,y)-γV(t,x,y), kZ, where γ>0 is a constant.

Moreover, one assumes that system (7) has a solution that remains in a compact set SD. Then, system (7) has a unique almost periodic solution which is uniformly asymptotically stable.

3. Permanence

In this section, we establish a permanence result for system (5).

Lemma 5 (see [<xref ref-type="bibr" rid="B10">10</xref>]).

Assume that xPC(R) with points of discontinuity at t=τk and is left continuous at t=τk for kZ+ and (8)x˙(t)f(t,x(t)),tτk,x(τk+)Ik(x(τk)),kZ+, where fC(R×R,R), IkC(R,R), and Ik(x) is nondecreasing in x for kZ+. Let u*(t) be the maximal solution of the scalar impulsive differential equation as (9)u˙(t)=f(t,u(t)),tτk,u(τk+)=Ik(u(τk))0,kZ+,u(t0+)=u0 existing on [t0,). Then, x(t0+)u0 implies x(t)u*(t) for tt0.

Remark 6.

If inequalities (8) in Lemma 5 are reversed and u*(t) is the minimal solution of system (9) existing on [t0,), then x(t0+)u0 implies x(t)u*(t) for tt0.

Lemma 7.

Assume that a,b>0; then, the following impulsive logistic equation (10)x˙(t)=x(t)[a-bx(t)],tτk,Δx(τk+)=hkx(τk),kZ+ has a unique globally asymptotically stable positive almost periodic solution x* which can be expressed as follows: (11)αeξAbx*(t)=[b-tW(t,s)ds]-1aηb(1-e-aθ), where A is defined as that in Lemma 3, ξ:=lnsupkZ(1/(1+hk)), α:=a-ξA, θ:=infkZτk1, η:=infkZj=12(1/(1+hj+k)), and (12)W(t,s)={e-a(t-s),τk-1<s<t<τk;j=mk+111+hje-a(t-s),              τm-1<sτm<τk<tτk+1.

Proof.

Let u=1/x; then, system (10) changes to (13)du(t)dt=-au(t)+b,      tτk,Δu(τk+)=-hk1+hku(τk),      kZ+. We can easily obtain that system (13) has a unique almost periodic solution which can be expressed as follows: (14)u*(t)=b-tW(t,s)ds. Then, system (10) has a unique almost periodic solution x* which can be expressed by (11). By Lemma 3, we have (15)x*(t)[b-teξAe-α(t-s)  ds]-1=αeξAb. On the other hand, (16)x*(t)[bt-θtηe-a(t-s)  ds]-1=aηb(1-e-aθ).

Suppose that x(t) is another positive solution of system (10). Define a Lyapunov function as (17)V(t)=|lnx*(t)-lnx(t)|,tR. For tτk, kZ+, calculating the upper right derivative of V(t) along the solution of system (10), we have (18)D+V(t)=-b|x*(t)-x(t)|. For t=τk, kZ+, we have (19)V(τk+)=|lnx*(τk+)-lnx(τk+)|=|ln(1+hk)x*(τk)(1+hk)x(τk)|=|lnx*(τk)-lnx(τk)|=V(τk). Therefore, V is nonincreasing. Integrating (18) from 0 to t leads to (20)V(t)+b0t|x(s)-x*(s)|dsV(0)<+,t0; that is, (21)0+|x(s)-x*(s)|ds<+, which implies that (22)lims+|x(s)-x*(s)|=0. Thus, the almost periodic solution of system (10) is globally asymptotically stable. This completes the proof.

Let (23)ηi=infkZj=1211+hi(j+k),ξi=lnsupkZ11+hik,kkkki=1,2,3.

Proposition 8.

Every solution (x1,x2,x2)T of system (5) satisfies (24)limsuptxi(t)Mi,i=1,2,3, where M1, M2, and M3 are defined as those in (27), (32), and (35), respectively.

Proof.

From the first equation of system (5), we have (25)x˙1(t)x1(t)[r1u-a1lx1(t)],tτk,x1(τk+)=(1+h1k)x1(τk),kZ+.

Consider the following auxiliary system: (26)z˙1(t)=z1(t)[r1u-a1lz1(t)],tτk,z1(τk+)=(1+h1k)z1(τk),kZ+. By Lemma 5, x1(t)z1(t), where z1(t) is the solution of system (26) with z1(0+)=x1(0+). By Lemma 7, system (26) has a unique globally asymptotically stable positive almost periodic solution z1* which can be expressed as follows: (27)z1*(t)=[a1l-tW1(t,s)ds]-1[a1lt-θtW1(t,s)ds]-1r1uη1a1l(1-e-r1uθ)=M1, where (28)W1(t,s)={e-r1u(t-s),τk-1<s<t<τk;j=mk+111+h1je-r1u(t-s),              τm-1<sτm<τk<tτk+1.

Then, for any constant ϵ>0, there exists T1>0 such that x1(t)z1(t)<z1*(t)+ϵM1+ϵ for t>T1. So, (29)limsuptx1(t)M1. For any ϵ>0, there exists T2>0 such that (30)x1(t)M1+ϵfor  tT2. From the second equation of system (5), we have (31)x˙2(t)x2(t)[-r2l+b2u(M1+ϵ)m1-a2lx2(t)],kkkkkkkkkkkkkkkkKK  tτk,  t>T2,x2(τk+)=(1+h2k)x2(τk),kZ+. Similar to the above argument as that in (29), one has (32)limsuptx2(t)m1-1b2uM1-r2lη2a2l[1-e(r2l-m1-1b2uM1)θ]=M2. Then, there exists T3>T2 such that (33)x2(t)M2+ϵfor  tT3. By the third equation of system (5), we have (34)x˙3(t)x3(t)[-r3l+b4u(M2+ϵ)m2-a3lx3(t)],kkkkkkkkkkkkkkkkkLLtτk,t>T3,x3(τk+)=(1+h3k)x3(τk),      kZ+. Similar to the above argument as that in (32), we have (35)limsuptx3(t)m2-1b4uM2-r3lη3a3l[1-e(r3l-m2-1b4uM2)θ]=M3. This completes the proof.

Proposition 9.

Let N1, N2, and N3 be defined as those in (42)–(48), respectively. Then, every solution (x1,x2,x3)T of system (5) satisfies (36)liminftxi(t)Ni,i=1,2,3, if the following condition holds:

r1l>m1-1b1uM2+ξ1A, b2lN1/(m1+M1α1)>r2u+(b3uM3/m2)+ξ2A, and b4lN2/(m2+M2α2)>r3u+ξ3A.

Proof.

According to Proposition 8, there exist ϵ>0 and T4>0 such that (37)r1l-m1-1b1u(M2+ϵ)-ξ1A0,xi(t)Mi+ϵfor  tT4,i=1,2,3. From the first equation of system (5), we have (38)x˙1(t)x1(t)[r1l-a1ux1(t)-b1u(M2+ϵ)m1],kkkkkkkkkkkkkkkktτk,t>T4,x1(τk+)=(1+h1k)x1(τk),      kZ+.

Consider the following auxiliary system: (39)p˙1(t)=p1(t)[r1l-m1-1b1u(M2+ϵ)-a1ux1(t)],kkkkkkkkkkkkkkkkkkkktτk,t>T4,p1(τk+)=(1+h1k)p1(τk),      kZ+. By Remark 15, x1(t)p1(t) for t>T4, where p1(t) is the solution of system (39) with p1(T4+)=x1(T4+). By Lemma 7, system (39) has a unique globally asymptotically stable positive almost periodic solution p1* which can be expressed as follows: (40)p1*(t)=[a1u-tW2(t,s)ds]-1r1l-m1-1b1u(M2+ϵ)-ξ1Aeξ1Aa1u=N1(ϵ),(41)W2(t,s)={e-[r1l-m1-1b1u(M2+ϵ)-ξ1A](t-s),τk-1<s<t<τk;j=mk+111+h1j              ×e-[r1l-m1-1b1u(M2+ϵ)-ξ1A](t-s),τm-1<sτm<τk<tτk+1. Similar to the above argument as that in (29), we have liminftx1(t)N1(ϵ). By the arbitrariness of ϵ, it leads to (42)liminftx1(t)N1:=r1l-m1-1b1uM2-ξ1Aeξ1Aa1u. Then, there exist 0<ϵ1ϵ and T5>T4 such that (43)b2l(N1-ϵ1)m1+(M1+ϵ1)α1-r2u-b3u(M3+ϵ1)m2-ξ2A0,x1(t)N1-ϵ1for  tT5.

By the second equation of system (5), we have (44)x˙2(t)x2(t)[b2l(N1-ϵ1)m1+(M1+ϵ1)α1-r2u-b3u(M3+ϵ1)m2kkkkkk-a2lx2(t)b2l(N1-ϵ1)m1+(M1+ϵ1)α1]tτk,t>T2,x2(τk+)=(1+h2k)x2(τk),      kZ+. Similar to the above argument as that in (42), one has (45)liminftx2(t)[(b2lN1/(m1+M1α1))-r2u-(b3uM3/m2)-ξ2A]eξ2Aa2u=N2. Then, there exist 0<ϵ2ϵ1 and T6>T5 such that (46)b4l(N2-ϵ2)m2+(M2+ϵ2)α2-r3u-ξ3A0,x2(t)N2-ϵ2for  tT6.

In view of the third equation of system (5), we have (47)x˙3(t)x3(t)[b4l(N2-ϵ2)m2+(M2+ϵ2)α2-r3u-a3ux3(t)],      kkkkkkkkkkkkkkkkkkkkktτk,t>T2,x3(τk+)=(1+h3k)x3(τk),      kZ+. Similar to the above argument as that in (45), one has (48)liminftx3(t)[(b4lN2/(m2+M2α2))-r3u-ξ3A]eξ3Aa3u=N3. This completes the proof.

Remark 10.

In view of (H1) in Proposition 9, the values of impulse coefficients hik  (i=1,2,3) and the number of the impulse points τk in each interval of length 1 have negative effect on the permanence of system (5).

By Propositions 8 and 9, we have the following theorem.

Theorem 11.

Assume that (H1) holds; then, system (5) is permanent.

Remark 12.

When hik(i=1,2,3)0 in system (5), then Theorem 11 changes to the corresponding permanence result in Bai and Wang . So, Theorem 11 extends the corresponding result in Bai and Wang . Further, Theorem 11 gives the sufficient conditions for the permanence of system (5) with almost periodic impulsive perturbations. Therefore, Theorem 11 provides a possible method to study the permanence of the models with impulsive perturbations in biological populations.

Remark 13.

From the proof of Propositions 8 and 9, we know that, under the conditions of Theorem 11, set S={(x1,x2,x3)TR3:NixiMi,i=1,2,3} is an invariant set of system (5).

4. Almost Periodic Solution

The main result of this paper is concerned with the existence of a unique uniformly asymptotically stable positive almost periodic solution for system (5).

Let (49)c1(t)=maxN1xM1α1xα1-1M2b1(t)(m1+xα1)2,c2(t)=maxN1xM1|m1+(1-α1)xα1|b2(t)(m1+xα1)2,d1(t)=b1(t)m1+N1α1,d2(t)=maxN2xM2α2xα2-1M3b3(t)(m2+xα2)2,d3(t)=maxN2xM2|m2+(1-α2)xα2|b4(t)(m2+xα2)2,e(t)=b3(t)m2+N2α2,tR.

Theorem 14.

Assume that (H1) holds; suppose further that

there exist positive constants λ1, λ2, λ3, and μ such that (50)inftR[λ1a1(t)-λ1c1(t)-λ2c2(t)]>μ,inftR[λ3a3(t)-λ2e1(t)]>μ,inftR[λ2a2(t)-λ1d1(t)-λ2d2(t)-λ3d3(t)]>μ;

then, system (5) admits a unique positive almost periodic solution, which is uniformly asymptotically stable.

Proof.

Suppose that Z(t)=(z1(t),z2(t),z3(t))T and Z*(t)=(z1*(t),z2*(t),z3*(t))T are any two solutions of system (5). Consider the product system of system (5) as (51)z˙1(t)=z1(t)[r1(t)-a1(t)z1(t)-b1(t)z2(t)m1+z1α1(t)],z˙2(t)=z2(t)[-r2(t)+b2(t)z1(t)m1+z1α1(t)kkkkkkk-a2(t)z2(t)-b3(t)z3(t)m2+z2α2(t)],z˙3(t)=z3(t)[-r3(t)+b4(t)z2(t)m2+z2α2(t)-a3(t)z3(t)],z˙1*(t)=z1*(t)[r1(t)-a1(t)z1*(t)-b1(t)z2*(t)m1+z1*α1(t)],z˙2*(t)=z2*(t)[-r2(t)+b2(t)z1*(t)m1+z1*α1(t)-a2(t)z2*(t)ggggkhhhk-b3(t)z3*(t)m2+z2*α2(t)],z˙3*(t)=z3*(t)[-r3(t)+b4(t)z2*(t)m2+z2*α2(t)-a3(t)z3*(t)],  kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkktτk,Δz1(τk)=h1kz1(τk),Δz2(τk)=h2kz2(τk),Δz3(τk)=h3kz3(τk),Δz1*(τk)=h1kz1*(τk),Δz2*(τk)=h2kz2*(τk),Δz3*(τk)=h3kz3*(τk),      kZ.

Set S1={(z1,z2,z3)TR3:NiziMi,i=1,2,3}, which is an invariant set of system (51) directly from Remark 13.

Construct a Lyapunov functional V(t,Z,Z*)=V(t,(z1,z2,z3)T,(z1*,z2*,z3*)T) defined on R+×S1×S1×S1 as follows: (52)V(t,Z,Z*)=i=13λi|lnzi(t)-lnzi*(t)|.

It is obvious that (53)V(t,Z,Z*)min{λ1,λ2,λ3}i=13|lnzi(t)-lnzi*(t)|min{λ1,λ2,λ3}i=131Mi|zi(t)-zi*(t)|λ_Z-Z*, where λ_=min{λ1,λ2,λ3}min{M1-1,M2-1,M3-1}. Further, we have (54)V(t,Z,Z*)max{λ1,λ2,λ3}i=13|lnzi(t)-lnzi*(t)|max{λ1,λ2,λ3}i=131Ni|zi(t)-zi*(t)|λ¯Z-Z*, where λ¯:=max{λ1,λ2,λ3}max{N1-1,N2-1,N3-1}; thus, (1) in Lemma 4 is satisfied.

Since (55)|V(t,Z,Z*)-V(t,Z¯,Z¯*)|=i=13λi|lnzi(t)-lnzi*(t)|-i=13λi|lnz¯i(t)-lnz¯i*(t)|λ¯i=13[|zi(t)-zi*(t)|+|z¯i(t)-z¯i*(t)|]=λ¯[|Z(t)-Z*(t)|+|Z¯(t)-Z¯*(t)|], (2) in Lemma 4 holds.

For tτk, kZ+, calculating the upper right derivative of V(t) along the solution of system (51), we have (56)D+V(t)=i=13λi[z˙i(t)zi(t)-z˙i*(t)zi*(t)]sgn(zi(t)-zi*(t))=λ1sgn(z1(t)kkkkkkk-z1*(t)){-a1(t)[z1(t)-z1*(t)]b1(t)z2*(t)m1+z1*α1(t)kkkkkkkkkkjjjjk-b1(t)z2(t)m1+z1α1(t)+b1(t)z2*(t)m1+z1*α1(t)}+λ2sgn(z2(t)-z2*(t))×{-a2(t)[z2(t)-z2*(t)]b3(t)z3*(t)m2+z2*α2(t)+b2(t)z1(t)m1+z1α1(t)-b2(t)z1*(t)m1+z1*α1(t)-b3(t)z3(t)m2+z2α2(t)+b3(t)z3*(t)m2+z2*α2(t)}+λ3sgn(z3(t)-z3*(t))×{-a3(t)[z3(t)-z3*(t)]b4(t)z2(t)m2+z2α2(t)+b4(t)z2(t)m2+z2α2(t)-b4(t)z2*(t)m2+z2*α2(t)}-[λ1a1(t)-λ1c1(t)-λ2c2(t)]|z1(t)-z1*(t)|-[λ2a2(t)-λ1d1(t)-λ2d2(t)-λ3d3(t)]×|z2(t)-z2*(t)|-[λ3a3(t)-λ2e1(t)]|z3(t)-z3*(t)|-i=13μλiNiλi|lnzi(t)-lnzi*(t)|-min{μλ1N1,μλ2N2,μλ3N3}V(t,Z,Z*).   For t=τk, kZ+, we have (57)V(τk+,Z(τk+),Z*(τk+))=i=13λi|lnzi(τk+)-lnzi*(τk+)|=i=13λi|ln(1+hik)zi(τk)(1+hik)zi*(τk)|=i=13λi|lnzi(τk)-lnzi*(τk)|=V(τk,Z(τk),Z*(τk)). In view of (56)-(57), (3) in Lemma 4 is satisfied.

By Lemma 4, system (5) admits a unique uniformly asymptotically stable positive almost periodic solution (z1(t),z2(t),z3(t))T. This completes the proof.

Remark 15.

When hik(i=1,2,3)0 in system (5), then Theorem 14 changes to the corresponding permanence result in Bai and Wang . So, Theorem 14 extends the corresponding result in Bai and Wang . Further, Theorem 14 gives the sufficient conditions for the uniform asymptotical stability of a unique positive almost periodic solution of system (5), in which α1 and α2 are allowed to be any real-valued positive number. Therefore, Theorem 14 provides a possible method to study the existence, uniqueness, and stability of positive almost periodic solution of the models with impulsive perturbations in biological populations.

5. An Example and Numerical Simulations Example 1.

Consider the following food chain system with impulsive perturbations: (58)x˙1(t)=x1(t)[-0.1cos(3t)x2(t)1+x1(t)2+cos(2t)-10x1(t)KKKKK-0.1cos(3t)x2(t)1+x1(t)],x˙2(t)=x2(t)[0.1x3(t)1+x2(t)-0.1sin(2t)GGGGGG+(5+cos(2t))x1(t)m1+x1(t)HHHHH-10x2(t)-0.1x3(t)1+x2(t)],x˙3(t)=x3(t)[(6+sin(3t))x2(t)1+x2(t)-0.1cos(5t)+(6+sin(3t))x2(t)1+x2(t)-10x3(t)(6+sin(3t))x2(t)1+x2(t)],tτk,Δx1(τk)=0.1x1(τk),Δx2(τk)=0.2x2(τk),Δx3(τk)=0.3x3(τk),k{0,1,}=Z+,θ=infkZτk1=1; then, system (58) is permanent and admits a unique uniformly asymptotically stable positive almost periodic solution.

Proof.

Corresponding to system (2), r1(t)=2+cos(2t), r2(t)=0.1sin(2t), r3(t)=0.1cos(5t), a1=a2=a310, b1(t)=0.1cos(3t), b2(t)=5+cos(2t), b3(t)=0.1, b4(t)=6+sin(3t), m1=m2=α1=α2=1, h1k0.1, h2k0.2, h3k0.3, kZ+. Taking λ1=λ2=λ3=1, the result is easy to obtain from Theorems 11 and 14; we would omit it (see Figures 1, 2, 3, 4, 5, and 6). This completes the proof.

State variable x1 of Example 1.

State variable x2 of Example 1.

State variable x3 of Example 1.

Stability of state variable x1 of Example 1.

Stability of state variable x2 of Example 1.

Stability of state variable x3 of Example 1.

6. Conclusion

By using the comparison theorem and the Lyapunov method of the impulsive differential equations, sufficient conditions are obtained which guarantee the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of a food chain system with almost periodic impulsive perturbations. Proposition 9 and Theorem 14 imply that the values of impulse coefficients hik  (i=1,2,3) and the number of the impulse points τk in each interval of length 1 are harm for the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of the model. The main results obtained in this paper are completely new and the method used in this paper provides a possible method to study the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of the models with impulsive perturbations in biological populations.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the reviewers for their valuable comments and constructive suggestions, which considerably improve the presentation of this paper. This work was supported by the Scientific Research Fund of Yunnan Provincial Education Department.

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