This paper studies multispecies nonautonomous Lotka-Volterra
competitive systems with delays and fixed-time impulsive effects. The sufficient conditions of integrable form on the permanence of species are established.

1. Introduction

In this paper, we consider the nonautonomous n-species Lotka-Volterra type competitive systems with delays and impulses xi′(t)=xi(t)[ai(t)-bi(t)xi(t)-∑j=1naij(t)xj(t-τij(t))],t≠tk,xi(tk+)=hikxi(tk),i=1,2,…,n,k=1,2,…,
where xi(t) represents the population density of the ith species at time t, the functions ai(t), bi(t), aij(t), and τij(t)(i,j=1,2,…,n) are bounded and continuous functions defined on R+=[0,+∞), aij(t)≥0, bi(t)≥0, τij(t)≥0 for all t∈R+, and impulsive coefficients hik for any i=1,2,…,n and k=1,2,… are positive constants.

In particular, when the delays τij(t)≡0 for all t∈R+ and i,j=1,2,…,n, then the system (1.1) degenerate into the following nondelayed non-autonomous n-species Lotka-volterra systemxi′(t)=xi(t)[ai(t)-∑j=1nbij(t)xj(t)],t≠tk,xi(tk+)=hikxi(tk),i=1,2,…,n,k=1,2,…,
where bii(t)=bi(t)+aii(t) and bij(t)=aij(t) for i,j=1,2,…,n and i≠j. For system (1.2), the author establish some new sufficient condition on the permanence of species and global attractivity in [1].

As we well know, systems like (1.1) and (1.2) without impulses are very important in the models of multispecies populations dynamics. Many important results on the permanence, extinction, global asymptotical stability for the two species or multi-species non-autonomous Lotka-Volterra systems and their special cases of periodic and almost periodic systems can be found in [2–14] and the references therein.

However, owing to many natural and man-made factors (e.g., fire, flooding, crop-dusting, deforestation, hunting, harvesting, etc.), the intrinsic discipline of biological species or ecological environment usually undergoes some discrete changes of relatively short duration at some fixed times. Such sudden changes can often be characterized mathematically in the form of impulses. In the last decade, much work has been done on the ecosystem with impulsive(see [1, 15–21] and the reference therein). Specially, the following system is considered in [22]:xi′(t)=xi(t)[ai(t)-bii(t)xi(t)-∑j=1,j≠in∫-∞0kj(s)xj(t+s)ds],t≠tk,xi(tk+)=hikxi(tk),i=1,2,…,n,k=1,2,….
The author establish some new sufficient conditions on the permanence of species and global attractivity for system (1.3). However, the effect of discrete delays on the possibility of species survival has been an important subject in population biology. We find that infinite delays are considered in the system (1.3). In this paper, it is very meaningful that discrete delays are proposed in the impulsive system (1.1).

2. Preliminaries

Let τ=sup{τij(t),t≥0,i,j=1,2,…,n}. We define Cn[-τ,0] the Banach space of bounded continuous function ϕ:[-τ,0]→Rn with the supremum norm defined by:‖ϕ‖c=sup-τ≤s≤0|ϕ(s)|,
where ϕ=(ϕ1,ϕ2,…,ϕn), and |ϕ(s)|=∑i=1n|ϕi(s)|. Define C+n[-τ,0]={ϕ=(ϕ1,ϕ2,…,ϕn)∈Cn[-τ,0]:ϕi(s)≥0, and ϕi(0)≥0 for all s∈[-τ,0] and i=1,2,…,n}. Motivated by the biological background of system (1.1), we always assume that all solutions (x1(t),x2(t),…,xn(t)) of system (1.1) satisfy the following initial condition:xi(s)=ϕi(s)∀s∈[-τ,0],i=1,2,…,n,
where ϕ=(ϕ1,ϕ2,…,ϕn)∈C+n[-τ,0].

It is obvious that the solution (x1(t),x2(t),…,xn(t)) of system (1.1) with initial condition (2.2) is positive, that is, xi(t)>0(i=1,2,…,n) on the interval of the existence and piecewise continuous with points of discontinuity of the first kind tk(k∈N) at which it is left continuous, that is, the following relations are satisfied:xi(tk-)=xi(tk),xi(tk+)=hikxi(tk),i=1,2,…,n,k∈N.

For system (1.1), we introduce the following assumptions:

functions ai(t),bi(t),aij(t) and τij(t) are bounded continuous on [0,+∞], and bi(t), aij(t) and τij(t)(i,j=1,2,…,n) are nonnegative for all t≥0.

for each 1≤i≤n, there are positive constants ωi>0 such that
liminft→∞(∫tt+ωibi(s)ds)>0,
and the functions
hi(t,μ)=∑t≤tk<t+μlnhik
are bounded for all t∈R+ and μ∈[0,ωi].

First, we consider the following impulsive logistic systemx′(t)=x(t)[α(t)-β(t)x(t)],t≠tk,x(tk+)=hkx(tk),k=1,2,…,
where α(t) and β(t) are bounded and continuous functions defined on R+, β(t)≥0 for all t∈R+, and impulsive coefficients hk for any k=1,2,… are positive constants. We have the following results.

Lemma 2.1.

Suppose that there is a positive constant ω such that
liminft→∞(∫tt+wβ(s)ds)>0,liminft→∞(∫tt+wα(s)ds+∑t≤tk<t+ωlnhk)>0,
and function
h(t,μ)=∑t≤tk<t+ωlnhk
is bounded on t∈R+ and μ∈[0,ω]. Then we have

there exist positive constants m and M such that
m≤liminft→∞x(t)≤limsupt→∞x(t)≤M,
for any positive solution x(t) of system (2.6);

limt→∞(x(1)(t)-x(2)(t))=0 for any two positive solutions x(1)(t) and x(2)(t) of system (2.6).

The proof of Lemma 2.1 can be found as Lemma 2.1 in [1] by Hou et al.

On the assumption (H_{2}), we firstly have the following result.

Lemma 2.2.

If assumption (H2) holds, then there exist constants d>0 and D>0 such that for any t2≥t1≥0|∑t1≤tk<t2lnhik|≤d(t2-t1)+D,i=1,2,…,n.

The proof of Lemma 2.2 is simple, we hence omit it here.

3. Main Results

Let xi0(t) be some fixed positive solution of the following impulsive logistic systems as the subsystems of system (1.1):xi′(t)=xi(t)[ai(t)-bi(t)xi(t)],t≠tk,xi(tk+)=hikxi(tk),k=1,2,….
On the permanence of all species xi(i=1,2,…,n) for system (1.1), we have the following result.

Theorem 3.1.

Suppose that assumptions (H1)-(H2) hold. If there exist positive constants ωi such that for each 1≤i≤n:
liminft→∞(∫tt+ωi(ai(s)-∑j≠inaij(s)xj0(s-τij(s)))ds+∑t≤tk<t+ωilnhik)>0,
and the functions
hi(t,μ)=∑t≤tk<t+μlnhik
are bounded for all t∈R+ and μ∈[0,ωi]. Then the system (1.1) is permanent, that is, there are positive constants γ>0 and M>0 such that
γ≤liminft→∞xi(t)≤limsupt→∞xi(t)≤M,i=1,2,…,n,
for any positive solution x(t)=(x1(t),x2(t),…,xn(t)) of system (1.1).

Proof.

Let x(t)=(x1(t),x2(t),…,xn(t)) be any positive solution of system (1.1). We first prove that the components xi(i=1,2,…,n) of system (1.1) are bounded. From assumption (H_{1}) and the ith equation of system (1.1), we have
xi′(t)≤xi(t)[ai(t)-bi(t)xi(t)],t≠tk,xi(tk+)=hikxi(tk),k∈N.
by the comparison theorem of impulsive differential equation, we have
xi(t)≤yi(t),∀t≥0,
where yi(t) is the solution of (3.1) with initial value yi(0)=xi(0). From the condition (3.2), we directly have
liminft→∞(∫tt+ωiai(s)ds+∑t≤tk<t+ωilnhik)>0,i=1,2,…,n.
Hence, from conclusion (a) of Lemma 2.1, we can obtain a constant Mi1>0, and there is a Ti1>0 such that yi(t)<Mi1 for all t≥Ti1. Let M=max1≤i≤n{Mi1} and T1=max1≤i≤n{Ti1}, we have
xi(t)≤M,∀t≥T1,i=1,2,…,n.
Hence, we finally have
limsupt→∞x(t)≤M.

Next, we prove that there is a constant γ>0 such that
liminft→∞x(t)≥γ,i=1,2,…,n.
For any t1 and t2 directly from system (1.1), we have
xi(t1)=xi(t2)exp(∫t2t1[ai(t)-bi(t)xi(t)-∑j=1naij(t)xj(t-τij(t))]dt+∑t2≤tk≤t1lnhik).
From condition (3.2), we can choose constants 0<ε<1 small enough and T2>0 large enough such that
∫tt+ωi(ai(s)-[bi(s)+aii(s)]ε-∑j≠inaij(s)[xj0(s-τij(s))+ε])ds+∑t≤tk<t+ωilnhik>ε,
for all t≥T2 and i=1,2,…,n. Considering (3.5), by the comparison theorem of impulsive differential equation and the conclusion (b) of Lemma 2.1., we obtain for the above ε≥0 that there is a T3>T2 such that
xi(t)≤xi0(t)+ε∀t≥T3,i=1,2,…,n,
where xi0(t) is a globally uniformly attractive positive solution of system (3.1).

Claim 1.

There is a constant η>0 such that limsupt→∞xi(t)>η(i=1,2,…,n) for any positive solution x(t)=(x1(t),x2(t),…,xn(t)) of system (1.1). In fact, if Claim 1 is not true, then there is an integer k∈{1,2,…,n} and a positive solution x(t)=(x1(t),x2(t),…,xn(t)) of system (1.1) such that
limsupt→∞xk(t)<ε.
Hence, there is a constant T4>T3 such that
xk(t)<ε∀t≥T4.
On the other hand, by (3.13) there is a T5≥T4 such that
xi(t)≤xi0(t)+ε∀t≥T5,
where i=1,2,…,n and i≠k. By (3.11) and (3.16), we obtain
xk(t)=xk(T5+τ)exp(∑T5+τ≤tk≤t∫T5+τt[ak(s)-bk(s)xk(t)-∑j=1nakj(s)xj(s-τij(s))]ds+∑T5+τ≤tk≤tlnhkk)≥xk(T5+τ)exp(∑T5+τ≤tk≤t∫T5+τt[ak(s)-(bk(s)+akk(s))ε-∑j=1,j≠knaij(s)(xj0(s-τij(s))+ε)]ds+∑T5+τ≤tk≤tlnhkk),
for all t≥T5+τ. Thus, from (3.12) we finally obtain limt→∞xk(t)=∞, which lead to a contradiction.

Claim 2.

There is a constant γ>0 such that liminft→∞xi(t)>γ(i=1,2,…,n) for any positive solution of system (1.1).

If Claim 2 is not true, then there is an integer k∈{1,2,…,n} and a sequence of initial function {ϕm}⊂C+[-τ,0] such that
liminft→∞xk(t,ϕm)<ηm2∀m=1,2,…,
where constant η is given in Claim 1. By Claim 1, for every m there are two time sequences sq(m) and tq(m), satisfying:
0<s1(m)<t1(m)<s2(m)<t2(m)<⋯<sq(m)<tq(m)<⋯,limq→∞sq(m)=∞,
such that
xk(sq(m),ϕm)≥ηm,xk(tq(m),ϕm)≤ηm2,ηm2≤xk(t,ϕm)≤ηm∀t∈(sq(m),tq(m)).
From the above proof, there is a constant T(m)≥T2 such that xi(t,ϕm)<M(i=1,2,…,n) for all t≥T(m). Further, there is an integer K1(m)>0 such that sq(m)>T(m) for all q>K1(m). From (3.11) and lemma 2.2., we can obtain
xk(tq(m),ϕm)≥xk(sq(m),ϕm)exp(∫sq(m)tq(m)[∑j=1nak(s)-bk(s)M-∑j=1nakj(s)M]ds+∑sq(m)≤tk≤tq(m)lnhkk)≥xk(sq(m),ϕm)exp(-(r1+d)(tq(m)-sq(m))-D),
where r1=supt≥0{|ai(t)|+bi(t)M+∑j=1naij(t)M}. Consequently, from (3.20) we have
tq(m)-sq(m)≥lnm-Dr1+d∀q>K1(m).
By (3.12), there is a large enough P>0 such that for all t≥T2, a≥P and a∈[lwi,(l+1)wi) and i=1,2,…,n, then, we obtain
∫tt+a(ai(s)-[bi(s)+aii(s)]ε-∑j≠inaij(s)[xj0(s-τij(s))+ε])ds+∑t≤tk<t+alnhik=∫tt+lwi(ai(s)-[bi(s)+aii(s)]ε-∑j≠inaij(s)[xj0(s-τij(s))+ε])ds+∑t≤tk<t+lwilnhik+∫t+lwit+a(ai(s)-[bi(s)+aii(s)]ε-∑j≠inaij(s)[xj0(s-τij(s))+ε])ds+∑t+lwi≤tk<t+alnhik>lε-r2wi,
where r2=supt≥0{|ai(t)|+[bi(t)+aii(t)]ε+∑j≠inaij(s)[xj0(s-τij(s))+ε]}. So, we choose L=2+(r2wi/ε) such that for all l>L, we have
∫tt+a(ai(s)-[bi(s)+aii(s)]ε-∑j≠inaij(s)[xj0(s-τij(s))+ε])ds+∑t≤tk<t+alnhik>ε.
From (3.23), there is an integer N0 such that for any m>N0 and q>K1(m), we have
ηm<ε,tq(m)-sq(m)>2Q,
where constant Q>P+τ.

So, when m>N0 and q>K1(m), for any t∈[sq(m)+Q+τ,tq(m)], from (3.11), (3.21), (3.25), and (3.26) we can obtain
xk(tq(m),ϕm)=xk(sq(m)+Q+τ,ϕm)×exp(∑sq(m)+Q+τ≤tk≤tq(m)∫sq(m)+Q+τtq(m)[ak(t)-bk(t)xk(t,ϕm)-∑j=1nakj(t)xj0(t-τkj(t)),ϕm]dt+∑sq(m)+Q+τ≤tk≤tq(m)lnhkk)
Consequently, from (3.20) and (3.25) it follows
ηm2≥ηm2×exp(∑sq(m)+Q+τ≤tk≤tq(m)∫sq(m)+Q+τtq(m)[ak(t)-(bk(t)+akk(t))ε-∑j=1,j≠knakj(t)xj0(t-τkj(t))+ε]dt+∑sq(m)+Q+τ≤tk≤tq(m)lnhkk)>ηm2.
This leads to a contradiction. Therefore, Claim 2 is true. This completes the proof.

When system (1.1) degenerates into the periodic case, then we can assume that there is a constant ω>0 and an integer q>0 such that ai(t+ω)=ai(t), bi(t+ω)=bi(t), aij(t+ω)=aij(t), tk+q=tk+ω and hik+q=hik for all t∈R+, k=1,2,… and i,j=1,2,…,n. From Remarks2.3 and 2.4 in [1], we can see the fixed positive solution xj0 of system (3.1) can be chosen to be the ω-periodic solution of system (3.1). Therefore, as a consequence of Theorem 3.1. we have the following result.

Corollary 3.2.

Suppose that system (1.1) is ω-periodic and for each i=1,2,…,n,
∫0ωbi(s)ds>0,∫0ω(ai(s)-∑j≠inaij(s)xj0(s-τij(s)))ds+∑k=1qlnhik>0.
Then, system (1.1) is permanent.

4. Numerical Example

In this section, we will give an example to demonstrate the effectiveness of our main results. We consider the following two species competitive system with delays and impulses:x1′(t)=x1(t)[a1(t)-b1(t)x1(t)-a11(t)x1(t-τ11(t))-a12(t)x2(t-τ12(t))],x2′(t)=x2(t)[a2(t)-b2(t)x2(t)-a21(t)x1(t-τ21(t))-a22(t)x2(t-τ22(t))],t≠tkx1(tk+)=h1kx1(tk),x2(tk+)=h2kx2(tk),k=1,2,….
We take a1(t)=2, a2(t)=b1(t)=b2(t)=a11(t)=a12(t)=a22(t)=1, a21=1-|sin(π/2)t|, τij(t)=2, h1k=e-1, h2k=e, tk=k. Obviously, system (4.1) is periodic with period ω=2. For q=2, we have tk+q=tk+ω, h1k+q=h1k and h2k+q=h2k for all k=1,2,…. Consider the following impulsive logistic systems as the subsystems of system (4.1):x1′(t)=x1(t)(2-x1(t)),x2′(t)=x2(t)(1-x2(t)),t≠kx1(t+)=e-1x1(tk),x2(t+)=ex2(tk),t=k.
According to the formula in [1], we can obtain that subsystem (4.2) has a unique globally asymptotically stable positive 2-periodic solution (x10(t),x20(t)), which can be expressed in following form:x10(t)=2x10x10+(2-x10)e-2(t-k),t∈[k,k+1),k=0,1,2,…,x20(t)=x20x20+(1-x20)e-(t-k),t∈[k,k+1),k=0,1,2,…,
where x10=(2(e-0.2-e-2)/(1-e-2)) and x20=(e-e-1)/(1-e-1). Since∫0ω(a1(t)-a12(t)x20(t-τ12(t)))dt+∑k=1qlnh1k=2∫01(2-x20x20+(1-x20)e-(t-2))dt+∑k=12lnh1k≈1.5244,∫0ω(a2(t)-a21(t)x10(t-τ21(t)))dt+∑k=1qlnh2k=2∫01(1-(1-sinπ2t)2x10x10+(2-x10)e-2(t-2))dt+∑k=12lnh2k≈3.8398,
we obtain that all conditions in Corollary 3.2 for system (1.1) holds. Therefore, from Theorem 3.1. we see that system (1.1) is permanent (see Figure 1).

Time series of x1(t) and x2(t).

Acknowledgments

This paper was supported by the National Sciences Foundation of China (11071283), the Sciences Foundation of Shanxi (2009011005-3), the Young foundation of Shanxi province (no. 2011021001-1), research project supported by Shanxi Scholarship Council of China (2011-093), the Major Subject Foundation of Shanxi, and Doctoral Scientific Research fund of Xinjiang Medical University.

HouJ.TengZ.GaoS.Permanence and global stability for nonautonomous N-species Lotka-Valterra competitive system with impulsesAhmadS.Rama Mohana RaoM.Asymptotically periodic solutions of n-competing species problem with time delaysFreedmanH. I.RuanS. G.Uniform persistence in functional-differential equationsFreedmanH. I.WuJ. H.Periodic solutions of single-species models with periodic delaySeifertG.On a delay-differential equation for single specie population variationsHeX.-Z.GopalsamyK.Persistence, attractivity, and delay in facultative mutualismGopalsamyK.KuangY.LisenaB.Global attractivity in nonautonomous logistic equations with delayTengZ.LiZ.Permanence and asymptotic behavior of the N-species nonautonomous Lotka-Volterra competitive systemsAhmadS.LazerA. C.On a property of nonautonomous Lotka-Volterra competition modelMuroyaY.Permanence of nonautonomous Lotka-Volterra delay differential systemsTengZ.Persistence and stability in general nonautonomous single-species Kolmogorov systems with delaysWuJ.ZhaoX.-Q.Permanence and convergence in multi-species competition systems with delayLiuB.TengZ.LiuW.Dynamic behaviors of the periodic Lotka-Volterra competing system with impulsive perturbationsNieL.TengZ.HuL.PengJ.The dynamics of a Lotka-Volterra predator-prey model with state dependent impulsive harvest for predatorJinZ.MaoanH.GuihuaL.The persistence in a Lotka-Volterra competition systems with impulsiveJinZ.ZhienM.MaoanH.The existence of periodic solutions of the n-species Lotka-Volterra competition systems with impulsiveXiaY.yhxia@zjnu.cnGlobal analysis of an impulsive delayed Lotka-Volterra competition systemWangW.ShenJ.LuoZ.Partial survival and extinction in two competing species with impulsesNieL.PengJ.TengZ.HuL.Existence and stability of periodic solution of a Lotka-Volterra predator-prey model with state dependent impulsive effectsHuH.hhxiao1@126.comWangK.WuD.Permanence and global stability for nonautonomous N-species Lotka-Volterra competitive system with impulses and infinite delays