# Partial Extinction, Permanence, and Global Attractivity in Nonautonomousn-Species Gilpin-Ayala Competitive Systems with Impulses

Authors: Juan Hou; Hanhui Liu
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

The qualitative properties of general nonautonomousn-species Gilpin-Ayala competitive systems with impulsive effects are studied. Some new criteria on the permanence, extinction, and global attractivity of partial species are established by using the methods of inequalities estimate and Liapunov functions.


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## Body

## 1. Introduction

In [1], the general nonautonomous n-species Lotka-Volterra competitive systems with impulsive effects are investigated. By using the methods of inequalities estimate and constructing the suitable Liapunov functions, the sufficient conditions on the permanence of whole species and global attractivity of systems are established.In [2], the authors studied the following general nonautonomous n-species Lotka-Volterra competitive systems with impulsive perturbations:
(1.1)x˙i(t)=xi(t)[ai(t)-∑j=1nbij(t)xj(t)],t≠tk,xi(tk+)=hikxi(tk),k=1,2,…,i=1,2,…,n,
and got a series of criteria on the extinction of a part of n-species, the permanence of other part of n-species, and the global attractivity of the systems.In [3], a periodic n-species Gilpin-Ayala competition system with impulses is studied and obtain some useful behaviors of the system.In this paper, we investigate the general nonautonomousn-species Gilpin-Ayala competitive systems with impulsive effects. 						(1.2)x˙i(t)=xi(t)[ai(t)-∑j=1nbij(t)xjθij(t)],t≠tk,xi(tk+)=hikxi(tk),k=1,2,…,i=1,2,…,n,
where bi(t) and aij(t)(i,j=1,2,…,n) are defined on R+=[0,∞) and are bounded continuous functions, aij(t)≥0 for all t∈R+, θij and hik>0 are constants for all k=1,2,… and i,j=1,2,…,n.


## 2. Preliminaries

Firstly, we introduce the following assumption.Assumption H.
There is a positive constantω such that for each i=1,2,…,n(2.1)lim⁡inf⁡t→∞⁡∫tt+ωaii(s)ds>0,lim⁡inf⁡t→∞⁡(∫tt+ω‍bi(s)ds+∑t⩽tk⩽t+μln⁡⁡hik)>0,
and functions
(2.2)hi(t,μ)=∑t⩽tk⩽t+μln⁡⁡hik,i=1,2,…,n
are bounded on t∈R+ and 0⩽μ⩽ω.For eachi∈{1,2,…,n}, we consider the following logistic impulsive equation as the subsystem of system (1.2)
(2.3)x˙i(t)=xi(t)[bi(t)-aii(t)xiθii(t)],t≠tk,xi(tk+)=hikxi(tk),k=1,2,….
From the above assumption, we have the following results.Lemma 2.1.
Suppose that assumption H holds. Then we have the following:(1)
There exist positive constantsm and M such that
(2.4)m⩽liminf⁡t→∞⁡xi(t)⩽limsup⁡t→∞⁡xi(t)⩽M,
for any positive solution ui(t) of (2.3).(2)
limt→∞(xi(1)(t)-xi(2)(t))=0 for any two positive solutions xi(1)(t) and xi(2)(t) of (2.3).Proof.
From assumption H, there are positive constantsk1, k2, δ and T0 such that for all t⩾T0 we have
(2.5)∫tt+ω(bi(s)-aii(s)k1)ds+∑t≤tk<t+ωln⁡⁡hk<-δ,(2.6)∫tt+ω(bi(s)-aii(s)k2)ds+∑t≤tk<t+ωln⁡⁡hk>δ.
From the boundedness of function h(t,μ)=∑t≤tk<t+μ‍lnhk, there is a positive constant P such that for any t∈R+ and μ∈[0,ω)(2.7)|h(t,μ)|=|∑t≤tk<t+μln⁡⁡hk|<P.
Firstly, we prove that there is a constantM>0 such that
(2.8)limsup⁡t→∞⁡xi(t)<M,
for any positive solution xi(t) of system (2.3). In fact, for any positive solution xi(t) of system (2.3), we only need to consider the following three cases.Case I. There is a t0≥T0 such that x(t)≥k1′=k1θii for all t≥t0.Case II. There is a t0≥T0 such that x(t)≤k1′ for all t≥t0.Case III.x(t) is oscillatory about k1′ for all t≥T0.
We first consider Case I. Sincexi(t)≥k1′ for all t≥t0, then for t=t0+lω, where l≥0 is any positive integer, integrating system (2.3) from t0 to t, from (2.5) we have
(2.9)xi(t)=xi(t0)exp⁡⁡(∫t0t‍(bi(s)-aii(s)xiθii(s))ds+∑t0≤tk<t‍ln⁡hk)≤xi(t0)exp⁡⁡(∑t0+(l-1)ω≤tk<t0+lω‍ln⁡hk∫t0t0+ω(bi(s)-aii(s)k1)ds+∑t0≤tk<tln⁡⁡hk+⋯+∫t0+(l-1)ωt0+lω(bi(s)-aii(s)k1)ds+∑t0+(l-1)ω≤tk<t0+lω‍ln⁡hk)≤xi(t0)exp⁡⁡(-lδ).
Hence, xi(t)→0 as l→∞, which leads a contradiction.
Next, we consider Case III. From the oscillation ofxi(t) about k1′, we can choose two sequences {ρn} and {ρn*} satisfying T0<ρ1<ρ1*<⋯<ρn<ρn*<⋯ and limn→∞ρn=limn→∞ρn*=∞ such that
(2.10)xi(ρn)≤k1′,xi(ρn+)≥k1′,xi(ρn*)≥k1′,xi(ρn*+)≤k1′,xi(t)≥k1′,∀t∈(ρn,ρn*),xi(t)≤k1′,∀t∈(ρn*,ρn+1).
For any t≥T0, if t∈(ρn,ρn*] for some integer n, then we can choose integer l≥0 and constant 0≤μ1<ω such that t=ρn+lω+μ1. Since
(2.11)x˙i(t)≤xi(t)(bi(t)-aii(t)k1),∀t∈(ρn,ρn*),t≠tk,
integrating this inequality from ρn to t, by (2.5) and (2.7) we obtain
(2.12)xi(t)=xi(ρn)exp⁡⁡(∫ρnt‍(bi(s)-aii(s)xθii(s))ds+∑ρn≤tk<tln⁡⁡hk)≤k1′exp⁡(∑ρn+lω≤tk<ρn+lω+μ1ln⁡⁡hk∫ρnρn+ω(bi(s)-aii(s)k1)ds+∑ρn≤tk<ρn+ωln⁡⁡hk+⋯+∫ρn+lωρn+lω+μ1‍(bi(s)-aii(s)k1)ds+∑ρn+lω≤tk<ρn+lω+μ1ln⁡⁡hk)≤k1′exp⁡⁡(-lδ+∫ρn+lωρn+lω+μ1(bi(s)-aii(s)k1)ds+∑ρn+lω≤tk<ρn+lω+μ1ln⁡⁡hk)≤k1′exp⁡⁡(α1ω+P),
where α1=supt∈R+{|bi(t)|+aii(t)k1}. If there is an integer n such that t∈(ρn*,ρn+1], then we obviously have
(2.13)xi(t)≤k1′<k1′exp⁡⁡(α1ω+P).
Therefore, for Case III we always have
(2.14)xi(t)≤k1′exp⁡⁡(α1ω+P),∀t≥T0.
Lastly, if Case II holds, then we directly have(2.15)xi(t)≤k1′exp⁡⁡(α1ω+P),∀t≥T0.
Choose constant M=k1′exp(α1ω+P), then we see that (2.8) holds.
Secondly, a similar argument as in the proof of (2.8) we can prove that there is a constant m>0, such that
(2.16)liminf⁡t→∞⁡x(t)>m,
for any positive solution xi(t) of system (2.3). Conclusion (1.1) is proved.
Now, we prove conclusion (1.2). Let xi(1)(t) and xi(2)(t) be any two positive solutions of system (2.3). From conclusion (1.1), it follows that there are positive constants A and B such that
(2.17)A⩽xi(1)(t),xi(2)(t)⩽B,∀t≥0.
Choose Liapunov function as follows:
(2.18)V(t)=|ln⁡⁡xi(1)(t)-ln⁡⁡xi(2)(t)|.
For any k=1,2,…, we have
(2.19)V(tk+)=|ln⁡⁡(hkxi(1)(tk))-ln⁡⁡(hkxi(2)(tk))|=V(tk).
Hence, V(t) is continuous for all t∈R+ and from the Mean-Value Theorem we can obtain
(2.20)1B|xi(1)(t)-xi(2)(t)|⩽V(t)⩽1A|xi(1)(t)-xi(2)(t)|.
Calculating the upper right derivative of V(t), then from (2.20) we obtain
(2.21)D+V=sign⁡(xi(1)(t)-xi(2)(t))(x˙i(1)(t)xi(1)(t)-x˙i(2)(t)xi(2)(t))=-aii(t)|xi(1)θii(t)-xi(2)θii(t)|≤-aii(t)[θii]Aiiθ|xi(1)(t)-xi(2)(t)|≤-aii(t)[θii]AθiiV(t),t≠tk,k=1,2,…,
where [θii]≤θii is the integer part of θii.
From this, we further have for anyt>0(2.22)V(t)⩽V(0)exp⁡(-[θii]Aθii∫0t‍aii(s)ds).
From condition (2.5) we can obtain ∫0t‍aii(t)dt→∞ as t→∞. Hence, V(t)→0 as t→∞. Further from (2.20) we finally obtain limt→∞(xi(1)(t)-xi(2)(t))=0. Conclusion (1.2) is proved. This completes the proof of Lemma 2.1.Applying Lemma2.1 and the comparison theorem of impulsive differential equations, we easily prove the following result.Lemma 2.2.
Suppose that assumption H holds then there is a constantB>0 such that
(2.23)limsup⁡t→∞⁡xi(t)≤B,i=1,2,…,n,
for any positive solution x(t)=(x1(t),x2(t),…,xn(t)) of system (1.2).


## 3. Extinction

On the partial extinction of system (1.2), we have the following result.Theorem 3.1.
Suppose that assumption H holds. Letr be a given integer and 1≤r<n. If for any l>r there is a il<l such that for any j≤l(3.1)θilj=θlj,(3.2)limsup⁡t→∞⁡∫tt+ωbl(s)ds+∑t⩽tk<t+ωln⁡⁡hlk∫tt+ωbil(s)ds+∑t⩽tk<t+ωln⁡hilk<liminf⁡t→∞⁡alj(t)ailj(t),∀j≤l,
or
(3.3)liminf⁡t→∞⁡∫tt+ωbil(s)ds+∑t⩽tk<t+ωln⁡⁡hilk∫tt+ωbl(s)ds+∑t⩽tk<t+ωln⁡⁡hlk>limsup⁡t→∞⁡ailj(t)alj(t),∀j≤l,
then species xi(i=r+1,r+2,…,n) are extinction, that is, for any positive solution x(t)=(x1(t),x2(t),…,xn(t)) of system (1.2),
(3.4)lim⁡t→∞⁡xi(t)=0,i=r+1,r+2,…,n.Proof.
Firstly, from assumption H, that (2.7) still holds and there are constants η0>0 and T0>0 such that
(3.5)∫tt+ωbi(s)ds+∑t⩽tk<t+ωln⁡⁡hik≥η0,
for all t≥T0 and i=1,2,…,n.
We first provexn(t)→0 as t→∞. Without loss of generality, we assume that condition (3.2) holds. When condition (3.3) holds, a similar argument can be given. Since
(3.6)limsup⁡t→∞⁡∫tt+ωbn(s)ds+∑t⩽tk<t+ωln⁡⁡hnk∫tt+ωbp(s)ds+∑t⩽tk<t+ωln⁡⁡hpk<liminf⁡t→∞anj(t)apj(t),j=1,2,…,n,
where p=in. Hence, we can choose positive constants α, β, ε and Tn≥T0 such that
(3.7)∫tt+ωbn(s)ds+∑t⩽tk<t+ωln⁡⁡hnk∫tt+ωbp(s)ds+∑t⩽tk<t+ωln⁡⁡hpk<αβ-ε<αβ<anj(t)apj(t),
for all t≥Tn and j=1,2,…,n. Hence, from (3.5) we further obtain
(3.8)∫tt+ω(-αbp(s)+βbn(s))ds+β∑t≤tk<t+ωln⁡⁡hnk-α∑t≤tk<t+ωln⁡⁡hpk<-βε(∫tt+ωbp(s)ds+∑t≤tk<t+ωln⁡⁡hpk)≤-βεη0,(3.9)αapj(t)-βanj(t)=βapj[αβ-anj(t)apj(t)]<0,
for all t≥Tn and j=1,2,…,n.
Consider the Liapunov function as follows:(3.10)Vn(t)=(xp(t))-α(xn(t))β.
Calculating the derivative, and from (3.1), we can obtain for any t≥0(3.11)dVn(t)dt=Vn(t)[-α(bp(t)-∑j=1napj(t)xjθpj(t))+β(bn(t)-∑j=1nanj(t)xjθnj(t))]=Vn(t)[-αbp(t)+βbn(t)+∑j=1n(αapj(t)-βanj(t))xjθpj(t)],
for all t≠tk and
(3.12)Vn(tk+)=hpk-αhnkβVn(tk),
for all k=1,2,…. From (3.9), we further have
(3.13)dVn(t)dt≤Vn(t)(-αbp(t)+βbn(t)),t≥Tn,t≠tk,Vn(tk+)=hpk-αhnkβVn(tk),k=1,2,….
For any t>Tn, there is an integer qt≥0 such that t∈[Tn+qtω,Tn+(qt+1)ω]. Hence, by integrating (3.13) from Tn to t, we obtain
(3.14)Vn(t)≤Vn(Tn)exp⁡⁡(∫Tnt[-αbp(s)+βbn(s)]ds+∑Tn≤tk<tln⁡⁡(hpk-αhnkβ))=Vn(Tn)exp⁡{∑Tn+qtω≤tk<tln⁡⁡(hpk-αhnkβ)∫TnTn+ω[-αbp(s)+βbn(s)]ds+∑Tn≤tk<Tn+ωln⁡⁡(hpk-αhnkβ)+⋯+∫Tn+qtωt[-αbp(s)+βbn(s)]ds+∑Tn+qtω≤tk<tln⁡⁡(hpk-αhnkβ)}≤Mnexp⁡⁡(-εβη0qt),
where(3.15)Mn=Vn(Tn)exp⁡(ωsup⁡t≥0⁡{α|bp(t)|+β|bn(t)|}+(α+β)P).
Since qt→∞ as t→∞, it follows that from (3.14)
(3.16)Vn(t)⟶0ast⟶∞.
Since
(3.17)(xn(t))β=Vn(t)(xp(t))α,(hnkxn(tk))β=hpkαxp(tk)hpk-αhnkβVn(tk),
by the boundedness of x(t) on [0,∞) (see Lemma 2.2), we have
(3.18)xn(t)⟶0ast⟶∞.
For any integerl>r, assume that we have obtained xi(t)→0 as t→∞ for all i>l. Now, we prove that xl(t)→0 as t→∞. Suppose that condition (3.3) holds. When condition (3.2) holds, the argument is similar. Let i=il, by (3.1), we have θilj=θlj, then for j≤l, we have θij=θlj. Then we can choose positive constants λ, η, δ and Tl≥T0 such that
(3.19)∫tt+ωbq(s)ds+∑t⩽tk<t+ωln⁡⁡hqk∫tt+ωbl(s)ds+∑t⩽tk<t+ωln⁡⁡hlk>λη+δ>λη>aqj(t)alj(t),
for all t≥Tl, and j=1,2,…,l, where q=il.
Consider the Liapunov function as follows:(3.20)Vl(t)=(xq(t))-η(xl(t))λ.
By calculating, we obtain for any t≥0(3.21)dVl(t)dt=Vl(t)[-ηbq(t)+λbl(t)+∑j=1l(ηaqj(t)-λalj(t))xjθlj(t)+∑j=l+1nηaqj(t)xjθqj(t)-∑j=l+1nλalj(t)xjθlj(t)],
for all t≠tk and
(3.22)Vl(tk+)=hqk-ηhlkλVl(tk),
for all k=1,2,…. From (3.3) and (3.19), we have
(3.23)∫tt+ω(-ηbq(s)+λbl(s))ds+λ∑t≤tk<t+ωln⁡⁡hlk-η∑t≤tk<t+ωln⁡⁡hqk<-ηδ(∫tt+ωbl(s)ds+∑t≤tk<t+ωln⁡⁡hlk)≤-δηη0,(3.24)ηaqj(t)-λalj(t)<0,
for all t≥Tl and j=1,2,…,l. Hence, from (3.21), it follows that
(3.25)dVl(t)dt≤Vl(t)[-ηbq(t)+λbl(t)+∑j=1l(ηaqj(t)-λalj(t))xjθlj(t)+∑j=l+1nηaqj(t)xjθqj(t)-∑j=l+1nλalj(t)xjθlj(t)],t≥Tl,t≠tk,Vn(tk+)=hpk-αhnkβVn(tk),k=1,2,….
Sincexi(t)→0 as t→∞ for all i>l, by the boundedness of aij(t)(i,j=1,2,…,n) on [0,∞), we obtain
(3.26)lim⁡t→∞⁡∫tt+ω∑j=l+1n(ηaqj(s)xjθqj(s)-λalj(s))xjθlj(s)ds=0.
Hence, for any small ε>0, there is a Tl′>0, such that
(3.27)∫tt+ω∑j=l+1n(ηaqj(s)xjθqj(s)-λalj(s))xjθlj(s)ds<ε,t>Tl′.
Combining (3.23), it follows that there is enough large Tl*>max{Tl,Tl′} such that for all t≥Tl*,(3.28)∫tt+ω[-ηbq(s)+λbl(s)+∑j=l+1n(ηaqj(s)xjθqj(s)-∑j=l+1nλalj(s))xjθlj(s)]ds-η∑t≤tk<t+ωln⁡⁡hqk+λ∑t≤tk<t+ωln⁡⁡hlk≤-12δηη0,xi(t)≤δ∀i>l.
For any t>Tl*, we firstly choose an integer qt≥0 such that t∈(Tl*+qtω,Tl*+(qt+1)ω]. Integrating (3.25) from Tl* to t, then from (3.3) and (3.28), we have
(3.29)Vl(t)≤Vl(Tl*)exp⁡⁡{∫Tl*t[-ηbq(s)+λbl(s)+∑j=l+1n‍(ηaqj(s)xjθqj(s)-∑j=l+1nλalj(s))xjθlj(s)]ds+∑Tl*≤tk<tln⁡⁡(hqk-ηhlkλ)}=Vl(Tl*)exp⁡⁡{(∑Tl*≤tk<Tl*+ωln⁡⁡(hqk-ηhlkλ)∫Tl*Tl*+ω[-ηbq(s)+λbl(s)+∑j=l+1nηaqj(s)xjθqj(s)-∑j=l+1nλalj(s)xjθlj(s)]ds+∑Tl*≤tk<Tl*+ωln⁡⁡(hqk-ηhlkλ))+⋯+(∑Tl*+(qt-1)ω≤tk<Tl*+qtω‍ln⁡(hqk-ηhlkλ)∫Tl*+(qt-1)ωTl*+qtω‍[-ηbq(s)+λbl(s)+∑j=l+1n‍ηaqj(s)xjθqj(s)-∑j=l+1n‍λalj(s)xjθlj(s)]ds+∑Tl*+(qt-1)ω≤tk<Tl*+qtω‍ln⁡(hqk-ηhlkλ))+(∑Tl*+qtω≤tk<t‍ln⁡(hqk-ηhlkλ)∫Tl*+qtωt‍[-ηbq(s)+λbl(s)+∑j=l+1n‍ηaqj(s)xjθqj(s)-∑j=l+1n‍λalj(s)xjθlj(s)]ds+∑Tl*+qtω≤tk<t‍ln⁡(hqk-ηhlkλ))}≤Mlexp⁡⁡(-12δηη0qt),
where
(3.30)Ml=Vl(Tl*)exp⁡⁡{(ωsup⁡t≥0⁡{η|bq(t)|+λ|bl(t)|+∑j=l+1n(ηaqj(t)+λalj(t))δθlj}+(λ+η)P)},
Since qt→∞ as t→∞, we obtain from (3.29)
(3.31)Vl(t)⟶0ast⟶∞,
Since
(3.32)(xl(t))η=Vl(t)(xq(t))λ,(hlkxl(tk))η=hlkηhqk-λVl(tk)(hqkxq(tk))λ,
by the boundedness of x(t) on [0,∞), it follows that
(3.33)xl(t)⟶0ast⟶∞.
Finally, by the induction principle, we obtain that xi(t)→0 as t→∞ for all i>r. This completes the proof of Theorem 3.1.


## 4. Permanence

In this section, we study the permanence of partial speciesxi(t)(i=1,2,…,r) of system (1.2). We state and prove the following result.Theorem 4.1.
Suppose that all the conditions of Theorem3.1 hold. If for each i=1,2,…,r(4.1)liminf⁡t→∞⁡(∫tt+ω[bi(s)-∑j≠iraij(s)uj0θij(t)]ds+∑t≤tk<t+ωln⁡⁡hik)>0,
where ui0 is some fixed positive solution of (2.3), then species xi(i=1,2,…,r) are permanent, that is, there are positive constants m and M such that for any positive solution x(t)=(x1(t),x2(t),…,xn(t)) of system (1.2)
(4.2)m≤liminf⁡t→∞⁡xi(t)≤limsup⁡t→∞⁡xi(t)≤M,i=1,2,…,r.Proof.
From (4.1) and the boundedness of functions aij(t)(i,j=1,2,…,n) on R+, there are constants ε0>0 and T1>0 such that for any t≥T1 and i=1,2,…,r.
(4.3)∫tt+ω[bi(s)-∑j=1naij(s)ε0-∑j≠iraij(s)uj0θij(s)]ds+∑t≤tk<t+ωln⁡⁡hik>ε0.
For any i≤r, from system (1.2), we have
(4.4)dxi(t)dt≤xi(t)[bi(t)-∑j=1naij(t)xjθij(t)],≤xi(t)[ai(t)-bii(t)xiθii(t)],t≠tk,t≥0,xi(tk+)=hikxi(tk),k=1,2,…,
we have(4.5)xi(t)≤ui(t)∀t≥0,
where ui(t) is the solution of (2.3) with initial condition ui(0)≥xi(0). From Lemma 2.1 and Theorem 3.1, for the above constant ε0 there is a T2≥T1 such that for all t≥T2(4.6)xi(t)≤ui(t)≤ui0(t)+ε0,i=1,2,…,r,(4.7)xi(t)<ɛ0,i=r+1,r+2,⋯,n.
Let
(4.8)γi=sup⁡t≥0⁡{|bi(t)|+∑j=1naij(t)ε0+∑j≠iraij(t)uj0θij(t)},m=min⁡1≤i≤r⁡{ε0exp⁡⁡(-γiω-P)},
where constant P>0 is given in (2.7). Obviously, m>0 and m is independent of any positive solution of system (1.2).
Now, we prove that there is aT3≥T2 such that
(4.9)xi(t)≥m∀t≥T3,i=1,2,…,r.
We only need to consider the following three cases for each i=1,2,…,r.Case I. There is a t1≥T2 such that xi(t)≤ε0′=ε0θii for all t≥t1.Case II. There is a t2≥T2 such that xi(t)≥ε0′ for all t≥t2.Case III. xi(t) oscillates about ε0′ for all t≥T2.
For Case I, lett=t1+lω, where l≥0 is any integer. From (4.3)–(4.7) we obtain
(4.10)xi(t)=xi(t1)exp⁡⁡(∫t1t(bi(s)-aii(s)xiθii(s)-∑j≠inaij(s)xjθij(s))ds+∑t1≤tk<tln⁡⁡hik)≥xi(t1)exp⁡⁡(∑t1+(l-1)ω≤tk<t1+lωln⁡⁡hik∫t1t1+ω(bi(s)-∑j=1naij(s)ε0-∑j≠ir‍aij(s)uj0θij(s))ds+∑t1≤tk<t1+ωln⁡⁡hik+⋯+∫t1+(l-1)ωt1+lω(bi(s)-∑j=1naij(s)ε0-∑j≠iraij(s)uj0θij(s))ds+∑t1+(l-1)ω≤tk<t1+lωln⁡⁡hik)≥xi(t1)exp⁡⁡(lε0).
Therefore, xi(t)→∞ as l→∞ which leads to a contradiction.
For Case III, we choose two sequences{ρn} and {ρn*} satisfying T2≤ρ1<ρ1*<⋯<ρn<ρn*<⋯ and limn→∞ρn=limn→∞ρn*=∞ such that
(4.11)xi(ρn)≥ε0′,xi(ρn+)≤ε0′,xi(ρn*)≤ε0′,xi(ρn*+)≥ε0′,xi(t)≤ε0′∀t∈(ρn,ρn*),xi(t)≥ε0′∀t∈(ρn*,ρn+1).
For any t≥T2, if t∈(ρn,ρn*] for some integer n, then we can choose an integer l≥0 such that t=ρn+lω+νi, where νi∈[0,ω) is a constant. Since for any t∈(ρn,ρn*) from (4.6) and (4.7) we have
(4.12)x˙i(t)≥xi(t)(bi(t)-∑j=1naij(t)ε0-∑j≠irbij(t)uj0θij(t)),t≠tk.
Integrating this inequality from ρn to t, then from (4.7) and (3.28)-(3.29) we have
(4.13)xi(t)≥x(ρn)exp⁡⁡(∫ρnt(bi(s)-∑j=1naij(s)ε0-∑j≠iraij(s)uj0θij(s))ds+∑ρn≤tk<tln⁡⁡hik)≥ε0exp⁡(∑ρn+(l-1)ω≤tk<ρn+lωln⁡⁡hik∫ρnρn+ω‍(bi(s)-∑j=1n‍aij(s)ε0-∑j≠ir‍aij(s)uj0θij(s))ds+∑ρn≤tk<ρn+ωln⁡⁡hik+⋯+∫ρn+(l-1)ωρn+lω(bi(s)-∑j=1naij(s)ε0-∑j≠iraij(s)uj0θij(s))ds+∑ρn+(l-1)ω≤tk<ρn+lωln⁡⁡hik)+∫ρn+lωρn+lω+νi(bi(s)-∑j=1naij(s)ε0-∑j≠ir‍aij(s)uj0θij(s))ds+∑ρn+lω≤tk<ρn+lω+νiln⁡⁡hik≥ε0exp⁡⁡(-γiω-P).
If there exists an integer n such that t∈(ρn*,ρn+1], then we obviously have
(4.14)xi(t)≥ε0>ε0exp⁡⁡(-γiω-P).
This shows that for Case III we always have
(4.15)xi(t)≥ε0exp⁡⁡(-γiω-P),∀t≥T2.
Finally, if Case II holds, then fromxi(t)≥ɛ0′ for all t≥t1, we can directly obtain that (4.9) holds.
Therefore, from Lemma2.2 and (4.9), it follows that species xi(t)(i=1,2,…,r) are permanent. This proof of Theorem 4.1 is completed.


## 5. Global Attractivity

In this section, we further discuss the global attractivity of speciesxi(t)(i≤r). In order to obtain our results, we first consider the following subsystem which is composed of the species xi(t)(i≤r) of system (1.2) and for convenience of statement we use the variable ui(t)(i≤r) to denote the species of this subsystem,
(5.1)dui(t)dt=ui(t)[bi(t)-∑j=1raij(t)ujθij(t)],t≠tk,ui(tk+)=hikui(tk),i=1,2,…,r,k=1,2,….We need the following lemma.Lemma 5.1.
Suppose that assumption H and condition (4.1) of Theorem 4.1 hold. Then subsystem (5.1) is permanent.
Lemma5.1 can be proved by using the same method given in the proof of Theorem 4.1. We now state and prove the main result of this section.Theorem 5.2.
Suppose that all conditions of Theorem3.1 and Theorem 4.1 hold. If there are positive constants ρ, D and di(i=1,2,…,r) and nonnegative integrable function μ(t) defined on R+, satisfying ∫st‍μ(τ)dτ≥-D+ρ(t-s) for all t≥s≥0, such that
(5.2)diaii(t)-∑j≠irdjaji(t)≥μ(t),i=1,2,…,r,
for all t≥0, then for any positive solution x(t)=(x1(t),x2(t),…,xn(t)) of system (1.2) and any positive solution u(t)=(u1(t),u2(t),…,ur(t)) of subsystem (5.1)
(5.3)lim⁡t→∞⁡(xi(t)-ui(t))=0,i=1,2,…,r.Proof.
Letx(t)=(x1(t),x2(t),…,xn(t)) be a positive solution of system (1.2) and u(t)=(u1(t),u2(t),…,ur(t)) be a positive solution of subsystem (5.1). By Theorem 3.1, we have xi(t)→0 as t→∞ for all i>r. From Theorem 4.1 and Lemma 5.1, there are positive constants m and M such that
(5.4)m≤xi(t),ui(t)≤M,i=1,2,…,r,
for all t≥0. Choose the Liapunov function as follows:
(5.5)Vr(t)=∑i=1rdi|ln⁡⁡xi(t)-ln⁡⁡ui(t)|.
Since
(5.6)Vr(tk+)=∑i=1rdi|ln⁡⁡xi(tk+)-ln⁡⁡ui(tk+)|=∑i=1rdi|ln⁡⁡hikxi(tk)-ln⁡⁡hikui(tk)|=Vr(tk),
then V(t) is continuous for all t≥0. Calculating the upper right derivative of Vr(t), we have
(5.7)D+Vr(t)≤∑i=1rdi(-aii|xiθii(t)-uiθii(t)|+∑j≠iraij(t)|xjθij(t)-ujθij(t)|)+g(t)=-∑i=1r(diaii-∑j≠irdjaji(t))|xiθji(t)-uiθji(t)|+g(t),
for all t≥0, where
(5.8)g(t)=∑i=1rdi∑j=r+1naji(t)xjθji(t).
By (5.2), we have
(5.9)D+Vr(t)≤-μ(t)∑i=1r|xiθji(t)-uiθji(t)|+g(t),∀t≥0.
By (5.4), we further obtain
(5.10)D+Vr(t)≤-λμ(t)Vr(t)+g(t),∀t≥0,
where λ=min1≤i≤rdi-1m>0. Applying the comparison theorem and the variation of constants formula of first-order linear differential equation, we have
(5.11)Vr(t)≤e-∫0t‍λμ(s)ds(∫0tg(s)e∫0s‍λμ(τ)dτds+Vr(0)),
for all t≥0. Since g(t)→0 as t→∞, from the properties of function μ(t) and (5.11), it is not hard to obtain Vr(t)→0 as t→∞. That shows
(5.12)lim⁡t→∞⁡(xi(t)-ui(t))=0,i=1,2,…,r.
This completes the proof of Theorem 5.2.



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*Source: 150841-2012-07-29.xml*