# Permanence and Periodic Solutions for a Two-Patch Impulsive Migration PeriodicN-Species Lotka-Volterra Competitive System

Authors: Zijian Liu; Chenxue Yang
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

We study a two-patch impulsive migration periodicN-species Lotka-Volterra competitive system. Based on analysis method, inequality estimation, and Lyapunov function method, sufficient conditions for the permanence and existence of a unique globally stable positive periodic solution of the system are established. Some numerical examples are shown to verify our results and discuss the model further.


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## Body

## 1. Introduction

Owing to natural enemy, severe competition, seasonal alternative, or deterioration of the patch environment, species dispersal (or migration) in two or more patches becomes one of the most prevalent phenomena of nature. Generally speaking, species dispersal is mainly concluded as the following three types: (i) dispersal occurs at every time and happens simultaneously between any two patches, that is, continuously bidirectional dispersal; (ii) dispersal occurs at some fixed time and happens simultaneously between any two patches, that is, impulsively bidirectional dispersal; (iii) dispersal shows itself as a total migration form, that is, impulsively unilateral diffusion (or migration).Many empirical works and monographs on population dispersal system with type (i) have been done (see [1–6] and references cited therein). For example, in [3], Teng and Lu have investigated the following single-species nonautonomous dispersal model with delays: (1)dxitdt=xitait-bitxit-citxit-τt-∫-σ0kit,sxit+sds+∑j=1nDijtxjt-xit,where Dij(t) represents the dispersal rate from patch j to patch i at time t and the dispersal established in this model is continuous and bidirectional; that is, the dispersal occurs at every time and happens simultaneously between any two patches i and j. In recent years, some population dynamical models with impulsively bidirectional dispersal have been proposed and studied (see [7–10] and references cited therein). For instance, in [7], the authors studied the following autonomous impulsive diffusion single species model: (2)dY1tdt=r1Y1tln⁡k1Y1tdY2tdt=r2Y2tln⁡k2Y2tt≠nτ,ΔY1=d1Y2t-Y1t,ΔY2=d2Y1t-Y2t,t=nτ,where di(i=1,2) is the dispersal rate in the ith patch. The pulse diffusion occurs at every τ period (τ is a positive constant). Obviously, in this model, species Y inhabits, respectively, two patches before the pulse appears; when the time at the pulse comes, species Y in two patches disperses from one patch to another, that is, impulsively bidirectional dispersal.However, in all of these investigated dispersal models considered so far, there are few papers to consider the total impulsive migration system, that is, impulsively unilateral diffusion (type (iii)) system. Practically, in the real ecological system, with seasonal alternative, some kinds of birds or vegetarians will migrate from cold patches (or food resource poor patches) to warm patches (or food resource rich patches) in search for a better habitat to inhabit or breed; fish will go back from ocean to their birthplace to spawn and so on. Obviously, this kind of diffusing behavior exists extensively in the real world. Therefore, it is a very basic problem to research this kind of impulsive migration systems. Zhang et al. in [11] studied a single species model with logistic growth and dissymmetric impulse dispersal and obtained some very general, weak conditions for the permanence, extinction of these systems, existence, uniqueness, and global stability of positive periodic solutions by using analysis based on the theory of discrete dynamical systems. In our previous work [12, 13], a two-patch impulsive diffusion periodic single-species logistic model (see [12]) and a two-patch prey impulsive diffusion periodic predator-prey model (see [13]) have been proposed and studied and some interesting results have been established, respectively. In this paper, we will continue our study on the two-patch impulsive diffusion model to a N-species competitive system.Motivated by the above analysis, in this paper, we consider the following two-patch impulsive migration periodicN-species Lotka-Volterra competitive system: (3)x˙it=xita1it-∑j=1nb1ijtxjt,t∈τ2k,τ2k+1,xiτ2k+1=1-D1ixiτ2k+1-,x˙it=xita2it-∑j=1nb2ijtxjt,t∈τ2k+1,τ2k+2,xiτ2k+2=1-D2ixiτ2k+2-,i=1,2,…,n,k=0,1,2,…,where xi is the population density of the ith species; a1i(t) and a2i(t) represent the intrinsic growth rates of the ith species in patch 1 and in patch 2, respectively; b1ii(t) and b2ii(t) denote the intraspecific competition coefficients of the ith species in patch 1 and in patch 2, respectively; b1ij(t) and b2ij(t)(j≠i) are the interspecific competition coefficients between the ith species and the jth species in patch 1 and in patch 2, respectively. The species migration occurs at every pulse time τk+1(k=0,1,2,…), where τ0=0,τ1<τ2<⋯<τk<⋯ is sequence of positive numbers with limk→∞τk=+∞. We suppose that the system is composed of two patches. When t∈[τ2k,τ2k+1), all the species live in patch 1; because of the change of the environment, the populations will migrate to patch 2 and the migration loss is D1ix(τ2k+1-)(i=1,2,…,n); then the populations will live in patch 2 during the period t∈[τ2k+1,τ2k+2). When the environment changes again, all the populations will migrate back to the previous patch; here, the migration loss is D2ix(τ2k+2-).In this paper, we always assume the following:(H1)
Functionsa1i(t),a2i(t),b1ij(t), and b2ij(t)(i,j=1,2,…,n) are T-periodic continuous defined on R+=[0,∞),b1ij(t)⩾0 and b2ij(t)⩾0 for all t∈R+ and i,j=1,2,…,n.(H2)
Impulsive time sequence{τk} satisfies τk+2=τk+T for all k=0,1,2,…. Moreover, for all i=1,2,…,n,D1i∈[0,1) and D2i∈[0,1) are constants.In addition, we assume that the investigatedN species always migrate between the two patches almost simultaneously. We will establish some sufficient conditions for the permanence, extinction, and existence of a unique globally asymptotically stable positive periodic solution of the system. The methods used in this paper are inequality estimation and Lyapunov functions which are introduced in work [14] “the permanence and global stability for nonautonomous N-species Lotka-Volterra competitive system with impulses.”The organization of this paper is as follows. In Section2, as preliminary, an important lemma on the two-patch impulsive migration periodic single-species logistic model is introduced. In Section 3, sufficient conditions on the permanence and extinction of system (3) are established. In Section 4, conditions for the existence and global stability of the unique positive periodic solution are obtained. Finally, some examples and numerical simulations are proposed to illustrate the feasibility of our results and discuss the model further.


## 2. Preliminaries

In this section, as a preliminary we consider the following two-patch impulsive migration periodic single-species logistic system:(4)x˙t=xtα1t-β1txt,t∈τ2k,τ2k+1,xτ2k+1=1-D1xτ2k+1-,x˙t=xtα2t-β2txt,t∈τ2k+1,τ2k+2,xτ2k+2=1-D2xτ2k+2-,k=0,1,2,…,where α1(t),α2(t),β1(t), and β2(t) are ω-periodic continuous functions defined on R+, β1(t)⩾0, and β2(t)⩾0 for all t∈R+ and impulsive time sequence {τk} satisfies τk+2=τk+ω for all k=0,1,2,…. Moreover, D1∈[0,1) and D2∈[0,1) are constants. We have the following result.Lemma 1.
Letx(t) be any positive solution of system (4).
(
a
) If system (4) satisfies (5)∫0τ1β1tdt+∫τ1ωβ2tdt>0,(6)∫0τ1α1tdt+∫τ1ωα2tdt+ln⁡1-D1+ln⁡1-D2>0,then it has a unique globally attractively positive ω-periodic solution x∗(t); that is, (7)limt→∞xt-x∗t=0.
(
b
) If condition (6) is replaced by (8)∫0τ1α1tdt+∫τ1ωα2tdt+ln⁡1-D1+ln⁡1-D2⩽0and condition (5) is retained, then (9)limt→∞xt=0.Proof.
Due to the fact that the population dispersal is only restricted in two patches and shows itself as aggregate migration, we can rewrite system (4) as follows: (10)x˙t=-1k+12xtα1t-β1txt+-1k+1+12xtα2t-β2txt,t∈τk,τk+1,xτk+1=-1k+121-D1xτk+1-+-1k+1+121-D2xτk+1-,k=0,1,2,….
In order to prove proposition (a), firstly, we prove the permanence of system (4); that is, there exist two positive constants m and M such that for any positive solution x(t) of system (4) we always have (11)m<liminft→∞xt⩽limsupt→∞xt<M.
From conditions (5) and (6), there are positive constants e1, e2, and δ such that (12)∫0τ1α1t-β1te1dt+∫τ1ωα2t-β2te1dt+ln⁡1-D1+ln⁡1-D2<-δ,(13)∫0τ1α1t-β1te2dt+∫τ1ωα2t-β2te2dt+ln⁡1-D1+ln⁡1-D2>δ.
We first of all prove that there is a constantM>0 such that (14)limsupt→∞xt<Mfor any positive solution of system (4). In fact, for any positive solution of system (4), we only need to consider the following three cases.
Case  1. There is t0⩾0 such that x(t)⩾e1 for all t⩾t0.
Case  2. There is t0⩾0 such that x(t)⩽e1 for all t⩾t0.
Case  3. x(t) is oscillatory about e1 for all t⩾0.
We first consider Case1. Since x(t)⩾e1 for all t⩾t0, then for t=t0+lω, where l is any positive integer, integrating system (10) from t0 to t, by (12), we have (15)xt=xt0exp⁡∫t0t-1k+12α1s-β1sxs+-1k+1+12α2s-β2sxsds+∑t0⩽τk<tln⁡-1k+121-D1+-1k+1+121-D2⩽xt0exp⁡l∫0ω-1k+12α1s-β1se1+-1k+1+12α2s-β2se1ds+ln⁡1-D1+ln⁡1-D2=xt0exp⁡l∫0τ1α1s-β1se1ds+∫τ1ωα2s-β2se1ds+ln⁡1-D1+ln⁡1-D2⩽xt0exp⁡-lδ. Hence, x(t)→0 as l→∞, which leads to a contradiction.
Next, we consider Case3. Obviously, there is t1⩾0 such that x(t1)<e1. Then we prove that, for all t⩾t1, (16)xt⩽e1expα0ω,where α0=maxt∈[0,ω]{|α1(t)|+|α2(t)|+(β1(t)+β2(t))e1}. If (16) is not true, then there is t2>t1 such that (17)xt2>e1expα0ω.Furthermore, there exists t3∈(t1,t2] such that x(t3)=e1 and x(t)⩾e1 for all t∈[t3,t2]. Taking an integer p⩾0 such that t2∈[t3+pω,t3+(p+1)ω), then for all t∈[t3,t2] we have (18)x˙t⩽-1k+12xtα1t-β1te1+-1k+1+12xtα2t-β2te1,t∈τk,τk+1,xτk+1=-1k+121-D1xτk+1-+-1k+1+121-D2xτk+1-,k=0,1,2,…,and integrating this inequality from t3 to t2 we have (19)xt2⩽xt3exp⁡∫t3t2-1k+12α1s-β1se1+-1k+1+12α2s-β2se1ds+pln⁡1-D1+ln⁡1-D2⩽e1exp⁡-pδexp⁡∫t3+pωt2-1k+12α1s-β1se1+-1k+1+12α2s-β2se1ds⩽e1exp⁡α0ω,which contradicts with (17). This proves that (16) holds.
Lastly, if Case2 holds, then we directly have (20)xt⩽e1expα0ω∀t⩾t1.Choose constant M=e1exp⁡(α0ω)+1; then we see that (14) holds.
By a similar argument as in the proof of (14) we can prove that there is a constant m>0 such that (21)liminft→∞xt>m for any positive solution x(t) of system (4). Conclusion (11) is proved.
Now, we prove proposition (a). Letx(t) and x∗(t) be any two positive solutions of system (4). It follows from (11) that there are positive constants A and B such that (22)A⩽xt,x∗t⩽B,∀t⩾0.Choose Lyapunov function as follows: (23)Vt=ln⁡xt-ln⁡x∗t. For any k=0,1,2,…, we have (24)Vτk+1=ln⁡-1k+121-D1+-1k+1+121-D2xτk+1--ln-1k+121-D1+-1k+1+121-D2x∗τk+1-=Vτk+1-.Hence, V(t) is continuous for all t∈R+ and from the Mean-Value Theorem we can obtain (25)1Bxt-x∗t⩽Vt⩽1Axt-x∗t.Calculating the upper right derivative of V(t), then from (25) we obtain (26)D+Vt=signxt-x∗tx˙txt-x˙∗tx∗t=-1k+12-β1t+-1k+1+12-β2txt-x∗t⩽--1k+12β1t+-1k+1+12β2tAVt,t≠τk+1,k=0,1,2,….From this, we further have, for any t=pω+γ, where p⩾0 is an integer and 0⩽γ<ω is a constant, (27)Vt⩽V0exp⁡-A∫0t-1k+12β1s+-1k+1+12β2sds=V0exp⁡-A∫0pω​+∫pωt​-1k+12β1s+-1k+1+12β2sds⩽V0exp-Ap∫0τ1β1sds+∫τ1ωβ2sds⟶0,p⟶∞. Hence, V(t)→0 as t→∞. Further from (25) we obtain (28)limt→∞xt-x∗t=0.
Lastly, we prove that system (4) has a unique positive ω-periodic solution. Consider the sequence x∗(mω,x0∗). It is obviously bounded in the interval [A,B] for all m=0,1,…. Let x¯∗ be a limit point of this sequence, x¯∗=limn→∞x∗(mnω,x0∗). Then x∗(ω,x¯∗)=x¯∗. Indeed, since x∗(ω,x∗(mnω,x0∗))=x∗(mnω,x∗(ω,x0∗)) and x∗(mnω,x∗(ω,x0∗))-x∗(mnω,x0∗)→0 as mn→∞, we get (29)x∗ω,x¯∗-x¯∗A,B⩽x∗ω,x¯∗-x∗ω,x∗mnω,x0∗A,B+x∗ω,x∗mnω,x0∗-x∗mnω,x0∗A,B+x∗mnω,x0∗-x¯∗A,B⟶0,n⟶∞.The sequence x∗(mω,x0∗),m=1,2,…, has a unique limit point. On the contrary, let the sequence have two limit points x¯∗=limn→∞x∗(mnω,x0∗) and x~∗=limn→∞x∗(mnω,x0∗). Then, taking into account (28) and x~∗=x∗(mnω,x~∗), we have (30)x¯∗-x~∗A,B⩽x¯∗-x∗mnω,x0∗A,B+x∗mnω,x0∗-x~∗A,B⟶0,n⟶∞,and hence x¯∗=x~∗. The solution x∗(t,x¯∗) is the unique periodic solution of system (4). By (28), it is globally attractive. This completes the proof of proposition (a).
Now we prove proposition (b). From (5) and (8), for any constant η>0, there is a positive constant δ0 such that (31)∫0τ1α1t-β1tηdt+∫τ1ωα2t-β2tηdt+ln⁡1-D1+ln⁡1-D2<-δ0. From this, a similar argument as in the proof of (14), we can obtain (32)xt⩽ηexp⁡α0ω for all t large enough. Finally, from the arbitrariness of η, we obtain x(t)→0 as t→∞. Lemma 1 is proved.Remark 2.
In [12], to prove the globally attractively positive ω-periodic solution and the extinction of system (4), we required conditions β1(t)>0 and β2(t)>0 for all t∈R+ besides conditions (6) and (8). However, we improve the conditions β1(t)>0 and β2(t)>0 for all t∈R+ to ∫0τ1β1(t)dt+∫τ1ωβ2(t)dt>0 (condition (5)) in Lemma 1, which is superior to conditions given in [12].


## 3. Permanence and Extinction

We first discuss the permanence of all species of system (3). A similar analysis as system (4), system (3) can also be written as follows: (33)x˙it=-1k+12xita1it-∑j=1nb1ijtxjt+-1k+1+12xita2it-∑j=1nb2ijtxjt,t∈τk,τk+1,xiτk+1=-1k+121-D1ixiτk+1-+-1k+1+121-D2ixiτk+1-,i=1,2,…,n,k=0,1,2,….For each i=1,2,…,n, we consider the following two-patch impulsive migration systems as the subsystems of system (3): (34)x˙it=xita1it-b1iitxit,t∈τ2k,τ2k+1,xiτ2k+1=1-D1ixiτ2k+1-,x˙it=xita2it-b2iitxit,t∈τ2k+1,τ2k+2,xiτ2k+2=1-D2ixiτ2k+2-,k=0,1,2,….On the permanence of all species xi(i=1,2,…,n) for system (3) we have the following result.Theorem 3.
Assume that conditions(H1) and (H2) hold. Moreover, if (35)∫0τ1b1iitdt+∫τ1Tb2iitdt>0,(36)∫0τ1a1it-∑j≠inb1ijtxj∗tdt+∫τ1Ta2it-∑j≠inb2ijtxj∗tdt+ln⁡1-D1i+ln⁡1-D2i>0,then system (3) is permanent; that is, there are constants m>0 and M>0 such that (37)m<liminft→∞xit⩽limsupt→∞xit<M,i=1,2,…,n for any positive solution x(t)=(x1(t),x2(t),…,xn(t)) of system (3), where xi∗(t)(i=1,2,…,n) is the globally attractively positive T-periodic solution of system (34).Proof.
From condition (36) we directly have (38)∫0τ1a1itdt+∫τ1Ta2itdt+ln⁡1-D1i+ln⁡1-D2i>0,and by Lemma 1(a) we can obtain that xi∗(t) defined in Theorem 3 is existent and globally attractive. Therefore, for any positive solution xi(t) of system (34) and any constant ε>0, there exists Tε>0 such that (39)xit⩽xi∗t+ε∀t⩾Tε,i=1,2,…,n.
We firstly prove the ultimately upper boundedness of system (3). From conditions (35) and (36), there are constants ε0>0 small enough such that (40)∫0τ1a1it-b1iitε0-∑j≠inb1ijtxj∗t+ε0dt+∫τ1Ta2it-b2iitε0-∑j≠inb2ijtxj∗t+ε0dt+ln⁡1-D1i+ln⁡1-D2i>ε0for each i=1,2,…,n. Let x(t)=(x1(t),x2(t),…,xn(t)) be any positive solution of system (3). Since (41)x˙it⩽xita1it-b1iitxit,t∈τ2k,τ2k+1,xiτ2k+1=1-D1ixiτ2k+1-,x˙it⩽xita2it-b2iitxit,t∈τ2k+1,τ2k+2,xiτ2k+2=1-D2ixiτ2k+2-,k=0,1,2,…,by the comparison theorem of impulsive differential equations, we obtain (42)xit⩽uit∀t⩾0, where ui(t) is the positive solution of system (34) with initial condition ui(0)=xi(0). By taking ε=ε0 in (39), we can obtain that (43)xit⩽xi∗t+ε0∀t⩾Tε0,i=1,2,…,n.Choose a constant M=maxt∈[0,T]{xi∗(t)+ε0:i=1,2,…,n}; then M is independent of any positive solution of system (3). Obviously, we have xi(t)⩽M for all t⩾Tε0 and i=1,2,…,n.
Next, we prove that there is a constantm>0 such that (44)liminft→∞xit>m,i=1,2,…,n. We only need to consider the following three cases for each i=1,2,…,n.
Case  1. There is t4⩾Tε0 such that xi(t)⩽ε0 for all t⩾t4.
Case  2. There is t4⩾Tε0 such that xi(t)⩾ε0 for all t⩾t4.
Case  3. xi(t) is oscillatory about ε0 for all t⩾Tε0.
For Case1, since xi(t)⩽ε0 for all t⩾t4, then let t=t4+lT, where l is any positive integer; integrating system (33) from t4 to t, by (40) and (43) we have (45)xit=xit4exp⁡∫t4t-1k+12a1is-b1iisxis-∑j≠inb1ijsxjs+-1k+1+12a2is-b2iisxis-∑j≠inb2ijsxjsds+∑t4⩽τk<tln⁡-1k+121-D1i+-1k+1+121-D2i⩾xit4exp⁡l∫0τ1a1is-b1iisε0-∑j≠inb1ijsxj∗s+ε0ds+∫τ1Ta2is-b2iisε0-∑j≠inb2ijsxj∗s+ε0ds+ln⁡1-D1i+ln⁡1-D2i⩾xit4exp⁡lε0.Hence, xi(t)→0 as l→∞, which leads to a contradiction.
For Case3, obviously, there is t5⩾Tε0 such that xi(t5)>ε0. Then we prove that, for all t⩾t5, (46)xit⩾ε0exp⁡-γ0T1-D1i1-D2i,where(47)γ0=maxt∈0,Ta1it+a2it+b1iit+b2iitε0+∑j≠inb1ijtxj∗t+ε0+∑j≠inb2ijtxj∗t+ε0.
If (46) is not true, then there is t6>t5 such that (48)xit6<ε0exp⁡-γ0T1-D1i1-D2i.Moreover, there exists t7∈(t5,t6] such that (49)xit7⩽ε0,xit7-⩾ε0,xit⩽ε0∀t∈t7,t6. If t6=t7, t6 must be an impulsive time. Then there exists a positive integer k such that t6=τ2k or t6=τ2k+1; thus we have (50)xit6=xit6-1-D2i,t=τ2k,xit6-1-D1i,t=τ2k+1.From this we can obtain (51)xit6⩾ε01-D1i1-D2i>ε0exp⁡-γ0T1-D1i1-D2i, which contradicts with (48). If t6>t7, we can choose an integer p⩾0 such that t6∈[t7+pT,t7+(p+1)T); then we have for all t∈[t7,t6](52)x˙it⩾-1k+12xita1it-b1iitε0-∑j≠inb1ijtxj∗t+ε0+-1k+1+12xita2it-b2iitε0-∑j≠inb2ijtxj∗t+ε0,t∈τk,τk+1,xiτk+1=-1k+121-D1ixiτk+1-+-1k+1+121-D2ixiτk+1-,k=0,1,2,…,and integrating this inequality from t7 to t6 we have (53)xit6⩾xit7-exp⁡∫t7t6-1k+12a1is-b1iisε0-∑j≠inb1ijsxj∗s+ε0+-1k+1+12a2is-b2iisε0-∑j≠inb2ijsxj∗sε0ds1-D1i1-D2ip+1⩾ε0exp⁡pε0exp⁡∫t7+pTt6-1k+12a1is-b1iisε0-∑j≠inb1ijsxj∗s+ε0+-1k+1+12a2is-b2iisε0-∑j≠inb2ijsxj∗sε0ds1-D1i1-D2i⩾ε0exp⁡-γ0T1-D1i1-D2i,which contradicts with (48) too. This proves that (46) holds.
Lastly, if Case2 holds, then we directly have (54)xit⩾ε0⩾ε0exp⁡-γ0T1-D1i1-D2i∀t⩾t4. Let constant m=min1⩽i⩽n{ε0exp(-γ0T-1)(1-D1i)(1-D2i)}. Then m is independent of any positive solution of system (3) and we finally have (55)liminft→∞xit>m,i=1,2,…,n. This completes the proof of Theorem 3.Next, we study the extinction of all speciesxi for system (3); we have the following result.Theorem 4.
Assume that conditions(H1) and (H2) hold. Moreover, if (56)∫0τ1b1iitdt+∫τ1Tb2iitdt>0,(57)∫0τ1a1itdt+∫τ1Ta2itdt+ln⁡1-D1i+ln⁡1-D2i⩽0,then all species of system (3) are extinct; that is, (58)limt→∞xit=0,i=1,2,…,nfor any positive solution x(t)=(x1(t),x2(t),…,xn(t)) of system (3).Proof.
From system (3) we directly have (59)x˙it⩽xita1it-b1iitxit,t∈τ2k,τ2k+1,xiτ2k+1=1-D1ixiτ2k+1-,x˙it⩽xita2it-b2iitxit,t∈τ2k+1,τ2k+2,xiτ2k+2=1-D2ixiτ2k+2-,i=1,2,…,n,k=0,1,2,….Hence, for each i=1,2,…,n, we have xi(t)⩽ui(t) for all t⩾0, where ui(t) is the positive solution of system (34) with initial condition ui(0)=xi(0). According to conditions (56) and (57), by Lemma 1(b), we finally have (60)limt→∞xit⩽limt→∞uit=0,i=1,2,…,n for any positive solution x(t)=(x1(t),x2(t),…,xn(t)) of system (3). Theorem 4 is completed.


## 4. Periodic Solutions

In this section, we study the existence, uniqueness, and the global stability of the positive periodic solution of system (3).Letx(t)=(x1(t),x2(t),…,xn(t)) and x∗(t)=(x1∗(t),x2∗(t),…,xn∗(t)) be any two positive solutions of system (3). From Theorem 3, we can obtain that there are constants A>0 and B>0 such that (61)A⩽xit⩽B,A⩽xi∗t⩽B,∀t⩾0,i=1,2,…,n.Theorem 5.
Suppose that all the conditions of Theorem3 hold and there are constants ci>0(i=1,2,…,n) and a nonnegative continuous function g(t), satisfying (62)∫0∞gtdt=∞,such that (63)min⁡cib1iit-∑j≠incjb1jit,cib2iit-∑j≠incjb2jit⩾gt∀t⩾0,i=1,2,…,n.Then system (3) has a unique positive T-periodic solution x∗(t)=(x1∗(t),x2∗(t),…,xn∗(t)) which is globally attractive; that is, any positive solution x(t)=(x1(t),x2(t),…,xn(t)) of system (3) satisfies (64)limt→∞xit-xi∗t=0,i=1,2,…,n.Proof.
Choose Lyapunov function as follows:(65)Vt=∑i=1nciln⁡xit-ln⁡xi∗t. For any impulsive time τk+1(k=0,1,2,…), we have (66)Vτk+1=∑i=1nciln⁡-1k+121-D1i+-1k+1+121-D2ixiτk+1--ln⁡-1k+121-D1i+-1k+1+121-D2ixi∗τk+1-=∑i=1nciln⁡xiτk+1--ln⁡xi∗τk+1-=Vτk+1-. Hence, V(t) is continuous for all t∈R+. On the other hand, from (61) we can obtain that for each i=1,2,…,n and any t∈R+ and t≠τk+1(67)1Bxit-xi∗t⩽ln⁡xit-ln⁡xi∗t⩽1Axit-xi∗t.For any t∈R+ and t≠τk+1(k=0,1,2,…), calculating the upper right derivative of V(t), from (63) and (67) we obtain(68)D+Vt=∑i=1ncisign⁡xit-xi∗tx˙itxit-x˙i∗txi∗t=∑i=1ncisign⁡xit-xi∗t-b1iitxit-xi∗t-∑j≠inb1ijtxjt-xj∗t,t∈τ2k,τ2k+1,∑i=1ncisign⁡xit-xi∗t-b2iitxit-xi∗t-∑j≠inb2ijtxjt-xj∗t,t∈τ2k+1,τ2k+2⩽∑i=1n-cib1iit+∑j≠incjb1jitxit-xi∗t,t∈τ2k,τ2k+1,∑i=1n-cib2iit+∑j≠incjb2jitxit-xi∗t,t∈τ2k+1,τ2k+2⩽-∑i=1nmint∈R+cib1iit-∑j≠incjb1jit,cib2iit-∑j≠incjb2jitxit-xi∗t⩽-∑i=1ngtxit-xi∗t⩽-δgtVt,where δ=A/max1⩽i⩽n{ci}>0. From this, we further have for any t⩾0(69)Vt⩽V0exp⁡-δ∫0tgsds.Hence, it follows from (62) that V(t)→0 as t→∞. Therefore, from (61) we obtain (70)limt→∞xit-xi∗t=0,i=1,2,…,n.
Now let us consider the sequence(x1∗(mT,z0),x2∗(mT,z0),…,xn∗(mT,z0))=z(mT,z0), where m=1,2,… and z0=(x1∗(0),x2∗(0),…,xn∗(0)). It is compact in the domain [A,B]n since A⩽xi∗(t)⩽B for all t⩾0 and i=1,2,…,n. Let z¯ be a limit point of this sequence, z¯=limn→∞z(mnT,z0). Then z(T,z¯)=z¯. Indeed, since z(T,z(mnT,z0))=z(mnT,z(T,z0)) and z(mnT,z(T,z0))-z(mnT,z0)→0 as mn→∞, we get(71)zT,z¯-z¯A,Bn⩽zT,z¯-zT,zmnT,z0A,Bn+zT,zmnT,z0-zmnT,z0A,Bn+zmnT,z0-z¯A,Bn⟶0,n⟶∞.The sequence z(mT,z0),m=1,2,… has a unique limit point. On the contrary, let the sequence have two limit points z¯=limn→∞z(mnT,z0) and z~=limn→∞z(mnT,z0). Then, taking into account (70) and z~=z(mnT,z~), we have (72)z¯-z~A,Bn⩽z¯-zmnT,z0A,Bn+zmnT,z0-z~A,Bn⟶0,n⟶∞,and hence z¯=z~. The solution (x1∗(t,z¯),x2∗(t,z¯),…,xn∗(t,z¯)) is the unique periodic solution of system (3). By (70), it is globally attractive. This completes the proof of Theorem 5.


## 5. Numerical Simulation and Discussion

In this paper, we have investigated a class of two-patch impulsive migration periodicN-species Lotka-Volterra competitive system. By means of inequality estimation and Lyapunov functions, we have given the criteria for the permanence, extinction, and existence of the unique globally stable positive periodic solution of system (3).In order to testify the validity of our results, we consider the following two-patch impulsive migration periodic2-species competitive system: (73)x˙1t=x1ta11t-b111tx1t-b112tx2t,x˙2t=x2ta12t-b121tx1t-b122tx2t,t∈τ2k,τ2k+1,x1τ2k+1=1-D11x1τ2k+1-,x2τ2k+1=1-D12x2τ2k+1-,x˙1t=x1ta21t-b211tx1t-b212tx2t,x˙2t=x2ta22t-b221tx1t-b222tx2t,t∈τ2k+1,τ2k+2,x1τ2k+2=1-D21x1τ2k+2-,x2τ2k+2=1-D22x2τ2k+2-,k=0,1,2,….Corresponding to system (34), two subsystems of system (73) are taken as follows: (74)x˙it=xita1it-b1iitxit,t∈τ2k,τ2k+1,xiτ2k+1=1-D1ixiτ2k+1-,x˙it=xita2it-b2iitxit,t∈τ2k+1,τ2k+2,xiτ2k+2=1-D2ixiτ2k+2-,i=1,2,k=0,1,2,….In system (73), we take τ0=0,τm-τm-1=1 for all m=1,2,…. Hence, we have T=τ2=2. Moreover, we take D11=D12=0.2, D21=D22=0.3, a11t=1+sin2πt,a12t=1.4+0.55sin⁡πt,a21t=-0.2+sin2πt, a22t=-0.2+0.5sin⁡πt, b111t=0.2+0.4cos2πt, b112(t)=0.1sin2πt, b121(t)=0.2sin2πt, b122(t)=0.55+0.4cos⁡(πt), b211(t)=1+0.8cos⁡(πt), b212(t)=0.2+0.1sin⁡(πt), b221(t)=0.3+0.1cos⁡(πt),b222(t)=0.3+0.1cos⁡(πt). Obviously, (75)∫01b111tdt+∫12b211tdt=1.4546>0,∫01a11tdt+∫12a21tdt+ln⁡1-D11+ln⁡1-D21=1.4934>0,which guarantee that system (74) with i=1 has a globally attractively positive 2-periodic solution x1∗(t) from Lemma 1(a) and 0.5⩽x1∗(t)⩽2. See Figure 1(a). Similarly, we have (76)∫01b122tdt+∫12b222tdt=0.85>0,∫01a12tdt+∫12a22tdt+ln⁡1-D12+ln⁡1-D22=0.6520>0;that is, system (74) with i=2 also has a globally attractively positive 2-periodic solution x2∗(t) from Lemma 1(a) and 0.2≤x1∗(t)≤1.8. See Figure 1(b).Figure 1
Dynamical behavior of system (74) for i=1 (a) and i=2 (b), the unique globally attractively positive 2-periodic solutions x1∗(t) and x2∗(t). Here, we take initial values x110=3,x120=2,x130=1,x210=1.5,x220=1, and x230=0.5.
(a)
(b)Further, it is not difficult to verify that(77)∫0τ1a11t-b112tx2∗tdt+∫τ1Ta21t-b212tx2∗tdt+ln⁡1-D11+ln⁡1-D21⩾∫0τ1a11t-1.8b112tdt+∫τ1Ta21t-1.8b212tdt+ln⁡1-D11+ln⁡1-D21=1.1334>0,∫0τ1a12t-b121tx1∗tdt+∫τ1Ta22t-b221tx1∗tdt+ln⁡1-D12+ln⁡1-D22⩾∫0τ1a12t-2b121tdt+∫τ1Ta22t-2b221tdt+ln⁡1-D12+ln⁡1-D22=0.0974>0,which satisfy condition (36) of Theorem 3 for each i=1,2. Therefore, species x1 and x2 are permanent. The numerical simulation is given in Figure 2.Figure 2
Dynamical behavior of system (73). Obviously, species x1 and x2 are permanent and 2-periodic. Here, we take initial values x10=1 and x20=2.Meanwhile, if we choosec1=c2=1 in Theorem 5, we can verify that (78)min⁡c1b111t-c2b121t,c1b211t-c2b221t,c2b122t-c1b112t,c2b222t-c1b212t⩾0.05cos⁡2πt⩾0; then we can choose gt=0.05cos2πt. Therefore we have that g(t) is nonnegative and continuous and ∫0∞gt=∫0∞0.05cos2πt=∞. All conditions of Theorem 5 are satisfied. Hence, system (74) has a unique positive 2-periodic solution (x¯1,x¯2). See also Figure 2.However, if the survival environment of the two patches is austere, the intrinsic growth rates of the two species will decrease. Hence, if we takea11(t)=0.5+|sin(2πt)|,a21(t)=-0.2+sin(πt),a12(t)=0.4+0.55sin⁡(πt), and a22(t)=-0.2+0.5sin⁡(πt) and all other parameters are retained, then we obtain (79)∫01a11tdt+∫12a21tdt+ln⁡1-D11+ln⁡1-D21=-0.2798<0,∫01a12tdt+∫12a22tdt+ln⁡1-D12+ln⁡1-D22=-0.3480<0,which satisfy condition (57) of Theorem 4 (condition (56) is obvious). Hence, all species of system (73) will go extinct. See Figure 3(a).Figure 3
The extinction of the two speciesx1 and x2 of system (73). The extinction illustrated in (a) is caused by the austere survival environment of the two patches and the extinction illustrated in (b) is caused by the large loss during the migration (D11=0.7,D12=0.6, D21=0.5, and D22=0.6). Here, we take initial values x10=1 and x20=2.
(a)
(b)In addition, if the environment of the two patches is survivable, but the migration loss of the two species is large, that is, if we takeD11=0.7, D12=0.6, D21=0.5, and D22=0.6 and all other coefficients are unchanged, then we can verify that (80)∫01a11tdt+∫12a21tdt+ln⁡1-D11+ln⁡1-D21=-0.0470<0,∫01a12tdt+∫12a22tdt+ln⁡1-D12+ln⁡1-D22=-0.3776<0,and condition (57) of Theorem 4 satisfies. Therefore, the two species x1 and x2 of system (73) will also go extinct. See Figure 3(b). Meanwhile, if we fix a11t=1+sin2πt,a12t=1.4+0.55sin⁡πt,a21t=-0.2+sin2πt, and a22(t)=-0.2+0.5sin⁡(πt), let D11=D21=D1 and D12=D22=D2 denote (81)fD1=∫01a11tdt+∫12a21tdt+ln⁡1-D1+ln⁡1-D1,gD2=∫01a12tdt+∫12a22tdt+ln⁡1-D2+ln⁡1-D2,and then we have f(D1)⩽0 if 0.6453⩽D1<1 and g(D2)⩽0 if 0.4598⩽D2<1 (see Figure 4); that is, species x1 and x2 will go extinct if 0.6453⩽D1<1 and 0.4598⩽D2<1. This shows that the migration loss during the migration also plays a crucial role on the permanence and extinction of the two species.Figure 4
The trends off(D1) on D1 (a) and g(D2) on D2 (b). Obviously, f(D1)⩽0 if 0.6453⩽D1<1 and g(D2)⩽0 if 0.4598⩽D2<1.
(a)
(b)Remark 6.
In the course of the above discussion, we have established conditions that guarantee that the two species are permanent or extinct simultaneously. Hence, an interesting and important open problem is under what conditions one species is permanent and the other is extinct.Remark 7.
In all of the above discussion, we have established that if(82)γ1=∫0τ1a11t-b112tx2∗tdt+∫τ1Ta21t-b212tx2∗tdt+ln⁡1-D11+ln⁡1-D21>0,γ2=∫0τ1a12t-b121tx1∗tdt+∫τ1Ta22t-b221tx1∗tdt+ln⁡1-D12+ln⁡1-D22>0,the two species are permanent. And if (83)η1=∫0τ1a11tdt+∫τ1Ta21tdt+ln⁡1-D11+ln⁡1-D21⩽0,η2=∫0τ1a12tdt+∫τ1Ta22tdt+ln⁡1-D12+ln⁡1-D22⩽0,then the two species are extinct. Therefore, we have the following open problems.(O1)
Ifγ1<0<η1,γ2>0, and η2⩽0, what trends of all solutions of system (73) are.(O2)
Ifγ2<0<η2,γ1>0, and η1⩽0, what trends of all solutions of system (73) are.(O3)
Ifγ1<0<η1 and γ2<0<η2, what trends of all solutions of system (73) are.
Moreover, if we extend the two-species competitive system to our investigatedN-species competitive system, what results can be obtained under the similar cases, which are also interesting open problems.



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