# Permanence and Almost Periodic Solutions forN-Species Nonautonomous Lotka-Volterra Competitive Systems with Delays and Impulsive Perturbations on Time Scales

Authors: Xuxu Yu; Qiru Wang; Yuzhen Bai
Journal: Complexity
 (2018)

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## Abstract

We investigate a class of nonautonomousN-species Lotka-Volterra-type competitive systems with time delays and impulsive perturbations on time scales. By using comparison theorems of impulsive dynamic equations on time scales, we obtain sufficient conditions to guarantee the permanence of the system. Then based on the Massera-type theorem for impulsive dynamic equations on time scales, we establish existence and uniformly asymptotic stability of the unique positive almost periodic solution of the system. Finally, an example is employed to illustrate our main results.


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## Body

## 1. Introduction

The well-known Lotka-Volterra models concerning ecological population, epidemiology, economics modeling, or even neural networks have been extensively investigated in the literature. Many researchers have studied the dynamical characteristics such as stability, persistence, periodicity, or almost periodicity of various Lotka-Volterra systems (see [1–6]). In [6], the authors studied the existence of globally asymptotically stable periodic, quasiperiodic, or almost periodic solutions of the nonautonomous Lotka-Volterra system.Impulsive differential equations have become important in recent years in mathematical models of real processes, and they rise in phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics. There have been significant developments in impulse theory in recent years (see [7–9]), and the fundamental theory of impulsive differential equations can be seen in the monographs [10–12].In [7], the authors considered the following N-species nonautonomous competitive system with impulsive perturbations and time delays:
(1)ẋit=xitbit−aitxit−∑j=1naijtxjt−τijt−∑j=1n∫−σij0cijt,sxjt+sds,t≠tk,k∈ℕ,i=1,2,…,n,xitk+=hikxitk,t=tk,where t∈ℝ+≔0,∞,xt=x1t,x2t,…,xnt∈ℝ+n≔x=x1,x2,…,xn:xi≥0,i=1,2,…,n represents the density vector of N-species at time t and hik for each i=1,2,…,n and k∈ℕ are positive constants.The theory of calculus on time scales was initiated by Hilger [13] in 1988 to unify continuous and discrete analysis (see [14, 15] and references cited therein). Dynamic equations on time scales include differential and difference equations as special cases, and their qualitative analysis is of particular importance (see [16–22]). In [23, 24], the authors considered the existence and stability of periodic solutions for Lotka-Volterra systems. In [25], the authors investigated the permanence and almost periodic solutions for a single-species system with impulsive effects on time scales.To the best of our knowledge, there is little work considering the permanence and stability of almost periodic solutions for theN-species nonautonomous Lotka-Volterra competitive system with delays and impulsive perturbations on time scales. Motivated by the above works, in this paper, we consider the following N-species nonautonomous Lotka-Volterra competitive system with delays and impulsive perturbations on time scales:
(2)xiΔt=xitbit−ditxit−∑j=1naijtxjt−τjt−∑j=1n∫−σj0cijt,sxjt+sΔs,t≠tk,t∈t0,+∞T,xitk+=hikxitk,t=tk,k∈ℕ,i=1,2,…,n,where T is an almost periodic time scale, t0∈T, and t0>0;xt=x1t,x2t,…,xnt∈ℝ+n,σj is positive constant, and time delays τjt (j=1,2,…,n) are bounded functions; xitk+=limt→tk+xit represents the population density xi after the impulse point tk, and hik is the effect of the kth impulse on species xi; the other notations can be seen in Section 2.The rest of this paper is organized as follows. In Section2, some useful definitions and lemmas are presented. In Sections 3, we discuss the permanence of (2). In Section 4, we establish existence and uniformly asymptotic stability of the almost periodic solution of (2). In Section 5, a numerical example is given to illustrate the feasibility of our results. Finally, we provide a discussion section.


## 2. Preliminaries and Lemmas

In this section, we introduce some basic definitions and lemmas which are used in this paper.A time scaleT is an arbitrary nonempty closed subset of ℝ. For t∈T, the forward and backward jump operators σ,ρ:T→T and the forward graininess μ:T→ℝ+ are defined, respectively, by
(3)σt≔infs∈T:s>t,ρt≔sups∈T:s<t,μt=σt−t.A pointt∈T is called left-dense if t>infT and ρt=t, left-scattered if ρt<t, right-dense if t<supT and σt=t, and right-scattered if σt>t. If T has a left-scattered maximum m, then Tk≔T\m; otherwise, Tk≔T. If T has a right-scattered minimum m, then Tk≔T\m; otherwise, Tk≔T.A functionf:T→ℝ is called right-dense continuous or rd-continuous provided that it is continuous at right-dense points in T and its left-side limits exist (finite) at left-dense points in T. A function f:T→ℝ is called continuous if and only if it is both left-dense continuous and right-dense continuous.For a functionf:T→ℝ and t∈Tk, we define the delta derivative of f at t, denoted fΔt, to be the number (provided it exists) with the property that given any ε>0, there is a neighborhood U of t such that
(4)fσt−fs−fΔtσt−s≤εσt−s,foralls∈U.Iff is rd-continuous, then there is a function F such that FΔt=ft. In this case, we define
(5)∫abftΔt=Fb−Fa.The functionp is regressive if 1+μtpt≠0, for all t∈Tk. Define the regressive class of functions on Tk to be R=p:T→ℝ p is rd-continuous and regressive}. We define the set R+ of all positively regressive elements by R+=R+T,ℝ=p∈R:1+μtpt>0 for all t∈T.Ifp∈R, then we define the generalized exponential function by
(6)ept,s=exp∫stξμτpτ△τ,for s,t∈T, where the cylinder transformation
(7)ξhz=log1+hzh,ifh≠0,z,ifh=0.Definition 1 (see [14, 15]).
Ifp,q∈R, then we define a circle plus addition by
(8)p⊕qt≔pt+qt+ptqtμt,for all t∈Tk. For p∈R, define a circle minus p by
(9)⊖p≔−p1+μp.Lemma 1 (see [14, 15]).
Letp,q∈R and s,t,r∈T. Then
(i)e0t,s≡1 and ept,t≡1;
(ii)epσt,s=1+μtptept,s;
(iii)ept,s=1/eps,t=eps,t;
(iv)ept,repr,s=ept,s;
(v)ept,sΔ=ptept,s;
(vi) ifa,b,c∈T, then
(10)∫abepc,σtptΔt=epc,a−epc,b.Lemma 2 (see [14, 15]).
Letf,g be delta differentiable functions on T. Then
(i)ν1f+ν2gΔ=ν1fΔ+ν2gΔ, for any constants ν1,ν2;
(ii)fgΔt=fΔtgt+fσtgΔt=ftgΔt+fΔtgσt;
(iii) iff and fΔ are continuous, then ∫atft,sΔsΔ=fσt,t+∫atfΔt,sΔs;
(iv) ifftfσt≠0, then 1/f is differentiable at t with
(11)1f∆t=−fΔtftfσt.Lemma 3 (see [14, 15]).
Assumep∈R and t0∈T. If 1+μtpt>0 for t∈T, then ept,t0>0, for all t∈T.Throughout this paper, we useEn to denote ℝn or ℂn.Definition 2 (see [26]).
A time scaleT is called an almost-periodic time scale if
(12)Π≔τ∈ℝ:t±τ∈T,∀t∈T≠0.Definition 3 (see [26]).
LetT be an almost-periodic time scale. A function f∈CT,En is called an almost periodic function if the ε-translation set of f.
(13)Eε,f=τ∈Π:ft+τ−ft<ε,∀t∈T
is a relatively dense set inT, for all ε>0; that is, for any given ε>0, there exists a constant lε>0 such that each interval of length lε contains a τε∈Eε,f such that
(14)ft+τ−ft<ε,∀t∈T.
τ is called the ε-translation number of f and lε is called the inclusion length of Eε,f.Definition 4 (see [26]).
LetT be an almost periodic time scale. A function ft,x∈CT×D,En is called an almost periodic function in t uniformly for x∈D, if the ε-translation set of f.
(15)Eε,f,S=τ∈Π:ft+τ,x−ft,x<ε,∀t,x∈T×S
is a relatively dense set inT, for all ε>0 and for each compact subset S of D; that is, for any given ε>0 and each compact set S in D, there exists a constant lε,S>0 such that each interval of length lε,S contains a τε,S∈Eε,f,S such that
(16)ft+τ,x−ft,x<ε,∀t,x∈T×S.
τ is called the ε-translation number of f and lε,S is called the inclusion length of Eε,f,S.Lemma 4 (see [26]).
(i) Letft,gt∈CT,En be almost periodic functions, then ft±gt,ft·gt are also almost periodic functions. If inft∈T∣gt∣>0, then the quotient ft/gt is almost periodic too.
(ii) Letft,x,gt,x∈CT×D,En be almost periodic in t uniformly for x∈D, then ft,x±gt,x,ft,x·gt,x are also uniformly almost periodic functions. If inft,x∈T×S∣gt,x∣>0, where S⊂D be any compact set, then the quotient ft,x/gt,x is uniformly almost periodic too.Definition 5
Speciesxii=1,2,…,n of (2) is said to be permanent if there are positive constants m and M such that
(17)m≤liminft→∞xit≤limsupt→∞xit≤M,for any positive solution xt=x1t,x2t,…,xnt of system (2).Lemma 5 (see [9]).
Assume thatx∈PC1T,ℝ and
(18)xΔt≤≥ptxt+qt,t≠tk,t∈t0,+∞T,xtk+≤≥dkxtk+bk,t=tk,k∈ℕ,then for t≥t0≥0,
(19)xt≤≥xt0∏t0<tk<tdkept,t0+∑t0<tk<t∏t0<tj<tdjept,tkbk+∫t0t∏s<tk<tdkept,σsqsΔs,where PC1=y:0,∞T→ℝ which is rd-continuous except at tk,k=1,2,…, for which ytk−, ytk+,yΔtk−,yΔtk+ exist with ytk−=ytk,yΔtk−=yΔtk.Throughout this paper, we letT be an almost periodic time scale. Denote BCℝ,ℝn the set of all bounded continuous functions from ℝ to ℝn, with norm
(20)f=max1≤i≤nsupt∈Tfit.We can see thatBCℝ,ℝn,· is a Banach space.Consider the following equation:(21)xΔt=ft,x,t≠tk,t∈T+,Δxtk=Ikxtk,t=tk,k∈ℕ,where f:T+×SB→ℝ,SB=x∈ℝ:x<B,x=supt∈Txt; the functions Ik∈Cℝ,ℝ, k∈ℕ are almost periodic uniformly with respects to x∈SB and are Lipschitz continuous in x, ft,x is almost periodic in t uniformly for x∈SB and is continuous in x. The set of sequences tkj,tkj=tk+j−tk,k,j∈ℕ is uniformly almost periodic and infktk1=θ>0. The product systems of (21) are as follows:
(22)xΔt=ft,x,t≠tk,t∈T+,Δxtk=Ikxtk,t=tk,k∈ℕ,yΔt=ft,y,t≠tk,t∈T+,Δytk=Ikytk,t=tk,k∈ℕ.DefineV1=V:T+×SB×SB→ℝ+, V is rd-continuous in tk−1,tkT+×SB×SB and limt,x,y→tk,x0,y0,t>tkVt,x,y=Vtk+,x0,y0.Lemma 6 (see [25]).
Suppose that there exists a Lyapunov functionalVt,x,y∈V1 satisfying the following conditions:
Condition 1. ax−y≤Vt,x,y≤bx−y, where t,x,y∈T+×SB×SB,a,b∈κ,κ=a∈Cℝ+,ℝ+:a0=0andaisincreasing.
Condition 2. Vt,x,y−Vt,x1,y1≤Lx−x1+y−y1, where t,x,y∈T+×SB×SB,L>0 is a constant.
Condition 3. Vtk+,x+Ikx,y+Iky≤Vt,x,y,x,y∈SB,t=tk,k∈ℕ.
Condition 4.D+V2.1Δt,x,y≤−cVt,x,y, where c>0,−c∈R+,x,y∈SB,t≠tk,k∈ℕ.
Moreover, if there exists a solutionxt∈S of (21) for t∈T+, where S⊂SB is a compact set, then there exists a unique almost periodic solution pt∈S of (21), which is uniformly asymptotically stable. In particular, if ft,x is ω-periodic in t uniformly for x∈SB and there exists a positive integer q such that tk+q=tk+ω,Ik+qx=Ikx with tk∈T+, then pt is also periodic.
Letϑ1,ϑ2 be the two positive constants and tk be a time sequence satisfying −ϑ1<t1<t2<⋯<tn<⋯ and tk→+∞ as k→+∞;ϑ2=max1≤j≤nsupt∈t0,∞Tτjt denote ϑ=maxϑ1,ϑ2. Let ϕs:t0−ϑ,t0T→ℝ+n be a piecewise continuous function satisfying that ϕs is continuous for all s∈t0−ϑ,t0T and s≠tk, for any s=tk∈t0−ϑ,t0T, ϕtk−=lims→tk−ϕs=ϕtk and ϕtk+=lims→tk+ϕs exist.For system (2), we introduce the following assumptions:
(H1)
bit,dit,aijt,τjt are almost periodic functions with positive constants b¯i,b¯i,d¯i,d¯i, and a¯ij such that b¯i≤bit≤b¯i,d¯i≤dit≤d¯i,andaijt≤a¯ij, for all t∈t0,+∞T,i,j=1,2,…,n.
(H2)
hik is an almost periodic sequence, and there exist positive constants h and H such that 1≤h≤∏s<tk<thik≤H for all s<t∈t0,+∞T,k∈ℕ.
(H3)
The kernelcijt,s≥0:t0,+∞T×−σj,0T is uniformly almost periodic in t for s∈−σj,0T and Δ-integrable on −σj,0T with ∫−σj0cijt,sΔs≤cij, for all t∈t0,+∞T; here, cij is positive constant.
(H4)
The set of sequencetkj,tkj=tk+j−tk,k,j∈ℕ, is uniformly almost periodic and infktk1=θ>0.


## 3. Permanence

In this section, we will discuss the permanence of system (2). First, we present the existence of solutions of system (2).Lemma 7.
Assume that (H1) and (H2) hold, then system (2) has a solution for any xt0=x1t0,…,xnt0∈ℝ+n.Proof.
For anyxt0=x1t0,…,xnt0∈ℝ+n, let
(23)φi0t=xit0∏t0<tk<thik,φimt=xit0∏t0<tk<thik+∫t0t∏s<tk<thikFis,φm−1Δs,m=1,2,…,where Fs,φm−1=F1s,φm−1,…,Fns,φm−1 is defined as
(24)Fit,φm−1=φim−1tbit−ditφim−1t−∑j=1naijtφjm−1t−τjt−∑j=1n∫−σj0cijt,sφjm−1t+sΔs,i=1,2,…,n.
Clearly,Ft,φm−1 satisfies local Lipschitz conditions for variable φm−1. By calculation, we can get that
(25)Fit,φm−1≤φm−1b¯i.
By Lemma5, we have that φmt=φ1mt,…,φnmt satisfies the following system.
(26)xiΔt=Fit,φm−1,t≠tk,t∈t0,+∞T,xitk+=hikxitk,t=tk,i=1,2,…,n,k∈ℕ.
DenoteK=I×G be any compact subset of t0,+∞T×ℝn, then we can choose positive constants ω,LK, and MG such that
(27)t−t0≤ω,Ft,φ−Ft,ψ≤LKφ−ψ,φ≤MG,for all t,φ,t,ψ∈K. Next, we will show the uniformly convergence of the sequence φmt.
Note that(28)φi1t−φi0t=∫t0t∏s<tk<thikFis,φ0Δs≤H∫t0tΔs≤HMGb¯it−t0≤HMGb¯iω.Thus, φi1t−φi0t≤HMGb¯ωi=1,2,…,n, where b¯=max1≤i≤nb¯i.
Assume that whenm=n,φint−φin−1t≤HnMGb¯LKn−1t−t0n/n!≤HnMGb¯LKn−1ωn/n! holds, then
(29)φin+1t−φint=∫t0t∏s<tk<thikFis,φn−Fis,φn−1Δs≤H∫t0tFis,φn−Fis,φn−1Δs≤H∫t0tLKφn−φn−1Δs≤H∫t0tLKHnMGb¯LKn−1s−t0nn!Δs≤Hn+1MGb¯LKnt−t0n+1n+1!≤Hn+1MGb¯LKnωn+1n+1!.
By mathematical induction method, we have thatφnt−φn−1t≤HnMGb¯LKn−1ωn/n!,n∈ℕ.
Note thatφmt=φ0t+∑n=1mφnt−φn−1t. According to Weierstrass’s judgment method, the sequence φmt is uniformly convergent on K. Since Ft,φm−1 is Lipschitz continuous in φm−1, we have that the sequence Ft,φm is also uniformly convergent.
Let(30)φt≔limm→+∞φmt,Φt≔limm→+∞Ft,φm−1=limm→+∞φmtΔ,forallt≠tk.
Ifσt=t, then
(31)φitΔ=φit′=lims→tφis−φits−t=lims→tlimm→+∞φis−φims+φims−φimt+φimt−φits−t=lims→tΦit+θs−t=Φit.
Here,θ∈0,1 is a constant.
Ifσt>t, then
(32)φitΔ=φiσt−φitσt−t=limm→+∞φiσt−φimσt+φimσt−φimt+φimt−φitσt−t=limm→+∞φimtΔ=Φit.
From above, we haveφtΔ=limm→+∞φmtΔ. In view of (26) and the arbitrary K, we can see that φt is a solution of system (2).Lemma 8.
Assume that (H1) and (H2) hold. Let xt=x1t,…,xnt be any given solution of system (2), then there exist positive constants xi∗ for i=1,2,…,n such that limsupt→+∞xit≤xi∗,i=1,2,…,n.Proof.
From (H1), we can choose constants ki for i=1,2,…,n, such that b¯i−d¯iki<0 and b¯i−d¯iki∈R+. For any positive solution xt=x1t,…,xnt of system (2) and each index i∈1,2,…,n, we only need to consider the following three cases:
Case I. There is a ti′≥t0∈T such that xit>ki, for all t∈ti′,+∞T.
Case II. There is a ti′≥t0∈T such that xit≤ki, for all t∈ti′,+∞T.
Case III.xit is oscillatory about ki, for all t∈t0,+∞T.
We first consider Case I. In view of system (2), we have that
(33)xiΔt≤xitbit−ditki≤xitb¯i−d¯iki,t≠tk,t∈ti′,+∞T,xitk+=hikxitk,t=tk,k∈ℕ.
By Lemma5 and (H2), for all t∈ti′,+∞T, we obtain
(34)xit≤xi∏ti′<tk<thikeb¯i−d¯ikit,ti′≤xiHeb¯i−d¯ikit,ti′.
Thus,xit→0ast→+∞, which leads to a contradiction.
Next, we consider Case III. From the oscillation ofxit about ki, we can choose two sequences ξin and ξin∗ satisfying t0<ξi1<ξi1∗<⋯<ξin<ξin∗<⋯ and limn→+∞ξin=limn→+∞ξin∗=+∞ such that
(35)xiξin≤ki,xiξin+≥ki,xiξin∗≥ki,xiξin∗+≤ki,xit≥ki,forallt∈ξin,ξin∗Txit≤ki,forallt∈ξin∗,ξin+1T.
For allt∈t0,+∞T, if t∈ξin,ξin∗T for some integer n, then xit≥ki and
(36)xiΔt≤xitbit−ditki≤xitb¯i−d¯iki,t≠tk,xitk+=hikxitk,t=tk,k∈ℕ.
By Lemma5, for all t∈ξin,ξin∗T, we have that
(37)xit≤xiξin∏ξin<tk<thikeb¯i−d¯ikit,ξin≤xiξinHeb¯i−d¯ikit,ξin≤kiH.
Sincexit≤ki≤Hkiforallt∈ξin∗,ξin+1T, together with Case II, we always have limsupt→+∞xit≤Hki=xi∗,i=1,2,…,n.We make another assumption.(H5)
b¯i−∑j=1na¯ij+cijxi∗>0,i=1,2,…,n.Lemma 9.
Assume that (H1), (H2), and (H5) hold. Let xt=x1t,…,xnt be any given solution of system (2), then there exist positive constants xi∗, for i=1,2,…,n, such that liminft→+∞xit≥xi∗,i=1,2,…,n.Proof.
By Lemma8, there exists t′∈t0,+∞T such that xit≤xi∗,∀t∈t′,+∞T,i=1,2,…,n, for any solution xt=x1t,…,xnt of system (2). From (H5), we can choose positive constants ki′, for i=1,2,…,n, such that b¯i−d¯iki′−∑j=1na¯ij+cijxi∗>0. In view of system (2), we have that, if t≠tk and t∈t′,+∞T,
(38)xiΔt=xitbit−ditxit−∑j=1naijtxjt−τjt−∑j=1n∫−σj0cijt,sxjt+sΔs≥xitb¯i−d¯ixit−∑j=1na¯ijxj∗−∑j=1nxj∗∫−σj0cijt,sΔs≥xitb¯i−d¯ixit−∑j=1na¯ij+cijxj∗,i=1,…,n.
For eachi∈1,2,…,n, similarly, we only need to consider the following three cases:
Case I. There is a ti″≥t′∈T such that xit≥ki′, for all t∈ti″,+∞T.
Case II. There is a ti″≥t′∈T such that xit<ki′ for all t∈ti″,+∞T.
Case III.xit is oscillatory about ki′ for all t∈t′,+∞T.
We first consider Case II. In view of system (2), we have that for all t∈ti″,+∞T,
(39)xiΔt≥xitb¯i−d¯ixit−∑j=1na¯ij+cijxj∗≥xitb¯i−d¯iki′−∑j=1na¯ij+cijxj∗,t≠tk,xitk+=hikxitk,t=tk,k∈ℕ.
By Lemma5 and (H2), for all t∈ti′′,+∞T, we obtain that
(40)xit≤xiti′∏ti′<tk<thikeb¯i−d¯ikit,ti′≤xiti′Heb¯i−d¯ikit,ti′
Thus,xit→+∞ast→+∞, which leads to a contradiction.
Next, we consider Case III. From the oscillation ofxit about ki′, we can choose two sequences ζin and ζin∗ satisfying t′<ζi1<ζi1∗<⋯<ζin<ζin∗<⋯ and limn→+∞ζin=limn→+∞ζin∗=+∞ such that
(41)xiζin≥ki′,xiζin+≤ki′,xiζin∗≤ki′,xiζin∗+≥ki′,xit≤ki′,forallt∈ζin,ζin∗T,xit≥ki′,forallt∈ζin∗,ζin+1T.
For allt∈t′,+∞T, if t∈ζin,ζin∗T for some integer n, xit≤ki′, then we have
(42)xiΔt≥xitb¯i−d¯iki′−∑j=1na¯ij+cijxj∗,t≠tk,xitk+=hikxitk,t=tk,k∈ℕ.
By Lemma5, for all t∈ζin,ζin∗T, we have
(43)xit≥xiζin∏ζin<tk<thikeb¯i−d¯iki′−∑j=1na¯ij+cijxj∗t,ζin≥hxiζineb¯i−d¯iki′−∑j=1na¯ij+cijxj∗t,ζin,from which we can see that xit→+∞ as t→+∞, which is a contradiction.
From the above, we see that only Case I holds and then we always haveliminft→+∞xit≥ki′=xi∗,i=1,2,…,n.Remark 1.
Through the proofs of Lemmas8 and 9, we can see that the constants xi∗=Hki and xi∗=ki′i=1,2,…,n are independent of the solutions of system (2).From Lemmas7, 8, and 9, we establish the permanence of system (2).Theorem 1.
Assume that (H1), (H2), and (H5) hold, then system (2) is permanent.Remark 2.
In Lemma7, we have proved the existence of solutions of system (2). However, in most of the previous similar results, the authors did not.We denote byΛ the set of all solutions xt=x1t,…,xnt of system (2) satisfying xi∗≤xit≤xi∗,∀t∈t0,+∞T. We can see that Λ,· is a Banach space (where ∥·∥ is defined in Section 2).Theorem 2.
Assume that (H1), (H2), (H3), (H4), and (H5) hold, then the set Λ≠Ø.Proof.
By Lemma7, let xt=x1t,…,xnt be a solution of system (2). From Theorem 1, for any ε>0, there exists a t∗≥t0 such that
(44)xi∗−ε≤xit≤xi∗+ε,∀t∈t∗,+∞T,i=1,2,…,n.
By the almost periodicity ofbit,dit, aijt, τjt, and cijt,s, there exists a sequence η=ηp∈Π with ηp→+∞ as p→+∞, such that for t≠tk, we have
(45)bit+ηp→bit,dit+ηp→dit,aijt+ηp→aijt,cijt+ηp,s→cijt,s,τjt+ηp→τjt,asp→+∞,and there exists a subsequence kl of p with kl→+∞ as l→+∞, such that tkl→tk,hikl→hik, as l→+∞, i=1,2,…,n.
Denotexipt=xit+ηp, for t≥t∗−ηp. For any positive integer q, it is easy to see that there exists a sequence xipt:p>q such that the sequence xipt has subsequence, denote by xipt again, converging on any finite interval of t∈t0,+∞T, as p→+∞. Thus, there is a function yt defined on T such that
(46)xipt→yitasp→+∞,∀t∈t0,+∞T.
Since(47)xipΔt=xiptbit+ηp−dit+ηpxipt−∑j=1naijt+ηpxjt+ηp−τjt+ηp−∑j=in∫−σj0cijt+ηp,sxjt+ηp+sΔs,t≠tk−ηp,k∈ℕ,xiptk+=hiklxitk+ηp,t=tk−ηp,i=1,2,…,n.
Lettingp→+∞, we have
(48)yiΔt=yitbit−dityit−∑j=1naijtyjt−τjt−∑j=1n∫−σj0cijt,syjt+sΔs,t≠tk,k∈ℕ,yitk+=hikyitk,t=tk,i=1,2,…,n.
Easily, we can see thatyt=y1t,…,ynt is a solution of system (2) and
(49)xi∗−ε≤yit≤xi∗+ε,∀t∈t0,+∞T,i=1,2,…,n.
Sinceε is an arbitrary small positive number, it follows that xi∗≤yit≤xi∗, for all t∈t0,+∞T, i=1,2,…,n.


## 4. Almost Periodic Solutions

In this section, we shall establish the existence and uniformly asymptotic stability of the unique almost periodic solution to system (2).Theorem 3.
Assume that (H1), (H2), (H3), (H4), and (H5) hold and further that
(H6)
There are positive constantsαi, for i=1,2,…,n, such that A<B with −A,−B∈R+, and
(50)αixi∗aiit−∑j≠iαjxj∗ajit≥0,forallt∈t0,+∞T,αixi∗ciit,s−∑j≠iαjxj∗cjit,s≥0,forallt,s∈t0,+∞T×−σi,0T,where A=max1≤i≤nd¯ixi∗−b¯i+∑j=1na¯ij+cijxj∗,B=min1≤i≤nd¯ixi∗2/xi∗, xi∗=Hki,xi∗=ki′i=1,2,…,n are as in Remark 1.
System (2) has a unique almost periodic solution in the set Λ, which is uniformly asymptotically stable.Proof.
From Theorem2, we can see that system (2) has a bounded solution x⋆t=x1⋆t,…,xn⋆t with xi∗≤xi⋆t≤xi∗,∀t∈t0,+∞T,i=1,2,…,n.
Consider the product systems of (2).
(51)xiΔt=xitbit−ditxit−∑j=1naijtxjt−τjt−∑j=1n∫−σj0cijt,sxjt+sΔs,t≠tk,t∈t0,+∞T,xitk+=hikxitk,t=tk,k∈ℕ,i=1,2,…,n,yiΔt=yitbit−dityit−∑j=1naijtyit−τjt−∑j=1n∫−σj0cijt,syjt+sΔs,t≠tk,t∈t0,+∞T,yitk+=hikyitk,t=tk,k∈ℕ,i=1,2,…,n.
Define Lyapunov functionVt,x,y on T+×Λ×Λ by
(52)Vt,x,y=∑i=1nαi1xit−1yit.
It is easy to see that there exist two positive constantsC1 and C2 such that
(53)C1x−y≤Vt,x,y≤C2x−y.
Leta,b∈Cℝ+,ℝ+,ax=C1x,bx=C2x, then Condition 1 of Lemma 6 is satisfied.
Noting that(54)Vt,x,y−Vt,x′,y′=∑i=1nαi1xit−1yit−∑i=1nαi1xi′t−1xi′t≤∑i=1nαi1xit−1yit−1xi′t−1yi′t≤∑i=1nαi1xit−1xi′t+1yit−1yi′t=∑i=1nαixit−xi′txitxi′t+yit−yi′tyityi′t≤∑i=1nαixi∗2xit−xi′t+yit−yi′t≤Lx−x′+y−y′,where x,y,x′,y′∈Λ,L=∑i=1nαi/xi∗2. Then Condition 2 of Lemma 6 is satisfied.
On the other hand, fort=tkk∈ℕ,
(55)Vtk+,xtk+,ytk+=∑i=1nαi1xitk+−1yitk+=∑i=1nαihik1xitk−1yitk≤∑i=1nαi1xitk−1yitk=Vtk,xtk,ytk,which means that Condition 3 of Lemma 6 is also satisfied.
In view of system (51), we have that, for t≠tkk∈ℕ,
(56)D1.2+VΔt,x,y=∑i=1nαi1xit−1yitΔ=∑i=1nαisignyit−xit1xit−1yitΔ=∑i=1nαisignyit−xit−xiΔtxitxiσt+yiΔtyityiσt=∑i=1nαisignyit−xit−1xiσtbit−ditxit−∑j=1naijtxjt−τjt−∑j=1n∫−σj0cijt,sxjt+sΔs+1yiσtbit−dityit−∑i=1naijtyjt−τjt−∑j=1n∫−σj0cijt,syjt+sΔs=∑i=1nαisignyit−xitxiσtyiσtbitxiσt−yiσt−ditxiσtyit−xityiσt−∑j=1naijtxiσtyjt−τjt−xjt−τjtyiσt−∑j=1n∫−σj0cijt,sxiσtyjt+s−xjt+syiσtΔs=∑i=1nαixiσtyiσt−bitxiσt−yiσt−ditxiσtyit−xit−xitxiσt−yiσt−∑j=1naijtsignyit−xitxiσtyjt−τjt−xjt−τjt−xjt−τjtxiσt−yiσt−∑j=1n∫−σj0cijt,ssignyit−xitxiσtyjt+s−xjt+s−xjt+sxiσt−yiσtΔs≤∑i=1nαixiσtyiσt−bitxiσt−yiσt−ditxiσtyit−xit−xitxiσt−yiσt−aiitxiσtyit−τit−xit−τit+∑j≠iaijtxiσtyjt−τjt−xjt−τjt+∑j=1naijtxjt−τjtyiσt−xiσt−∫−σi0ciit,sxiσtyit+s−xit+sΔs+∑j≠i∫−σj0cijt,sxiσtyjt+s−xjt+s+∑j=1n∫−σj0cijt,sxjt+sΔsxiσt−yiσtΔs=∑i=1nαixiσtyiσtditxit−bit+∑j=1naijtxjt−τjt+∑j=1n∫−σj0cijt,sxjt+sΔsxiσt−yiσt−ditxiσtyit−xit−xiσtaiityit−τit−xit−τit−∑j≠iaijtyjt−τjt−xjt−τjt−xiσt∫−σi0ciityit+s−xit+sΔs−∑j≠i∫−σj0cijt,syjt+s−xjt+sΔs=∑i=1nαixiσtyiσtditxit−bit+∑j=1naijtxjt−τjt+∑j=1n∫−σj0cijt,sxjt+sΔsxiσt−yiσt−ditxiσtyit−xit−∑i=1nαixiσtxiσtyiσtaiityit−τit−xit−τit−∑j≠iaijtyjt−τjt−xjt−τjt−∑i=1nαixiσtxiσtyiσt×∫−σi0ciityit+s−xit+sΔs−∑j≠i∫−σj0cijt,syjt+s−xjt+sΔs=∑i=1nαixiσtyiσtditxit−bit+∑j=1naijtxjt−τjt+∑j=1n∫−σj0cijt,sxjt+sΔsxiσt−yiσt−ditxiσtyit−xit−∑i=1nαiyiσtaiityit−τit−xit−τit−∑j≠iαjyjσtajityit−τit−xit−τit−∑i=1n∫−σi0αiyiσtciit,syit+s−xit+sΔs−∑j≠i∫−σi0αjyjσtcjit,syit+s−xit+sΔs≤∑i=1nαixiσtyiσtditxit−bit+∑j=1naijtxjt−τjt+∑j=1n∫−σj0cijt,sxjt+sΔsxiσt−yiσt−ditxiσtyit−xit=∑i=1nαiditxit−bit+∑j=1naijtxjt−τjt+∑j=1n∫−σj0cijt,sxjt+sΔs1xiσt−1yiσt−∑i=1nαixityitdityiσt1xit−1yit≤∑i=1nαidi¯xi∗−bi¯+∑j=1naij¯+cijxj∗1xiσt−1yiσt−∑i=1nαidi¯xi∗2xi∗1xit−1yit≤AVσt,x,y−BVt,x,y=A−BVt,x,y+AμtD1.2+VΔt,x,y.
It follows that(57)D1.2+VΔt,x,y≤A−B1−Aμ¯Vt,x,y=−ηVt,x,y,where constant μ¯=supt∈t0,+∞Tμt. By (H6), we have that η=B−A/1−Aμ¯>0 and −η∈R+. Then Condition 4 of Lemma 6 holds. Hence, according to Lemma 6, there exists a unique uniformly asymptotically stable almost periodic solution of system (2) in the set Λ.


## 5. Applications

In order to validate our results, we consider a kind of three-species system with delays and impulsive perturbations on time scales as a special situation of system (2), where T=1/5ℤ.Chooset0=0,n=3, we let αj=σj=1,h=e1/π,H=e2,tk=k,
(58)hik=expsin1k2,τjt=2+sinjt,bit=6.6+0.1sin7it,dit=1.6+0.1cos5it,aiit=21000+11000cos6it,ciit,s=110001.8+0.2sin7ite1s,0,and for alli≠j,
(59)aijt=210000+110000sini+jt,cijt,s=1100001.8+0.2cosi+jte1s,0.By calculating, we have that(60)1<e1/π≤∏t0<tk<thik≤e2,b¯i=6.7,b¯i=6.5,d¯i=1.7,d¯i=1.5,a¯ii=31000,a¯ii=11000,cii=21000,foralli=1,2,3,and for all i≠j, we have
(61)a¯ij=310000,a¯ij=110000,cij=210000.Easily, we can see that (H1), (H2), (H3), and (H4) hold. Choosing constants xi∗=4.5,xi∗=3.8i=1,2,3, it follows that
(62)b¯i−d¯ixi∗=−0.05<0∈R+,b¯i−∑i=1na¯ij+cijxj∗=6.473>0,b¯i−d¯ixi∗¨−∑i=1na¯ij+cijxj∗=0.013>0,αixi∗aiit−∑j≠iαjxj∗ajit>0.0005>0,αixi∗ciit,s−∑j≠iαjxj∗cjit,s>0.0001>0,A=max1≤i≤3d¯ixi∗−b¯i+∑j=1na¯ij+cijxj∗=1.177,B=min1≤i≤3d¯ixi∗2xi∗=4.813,A<B.That is, (H5) and (H6), hold. Then, all conditions in Theorem 3 are satisfied. Thus, system (2) has a unique positive almost periodic solution, which is uniformly asymptotically stable.


## 6. Discussion

In this paper, we extend theN-species nonautonomous Lotka-Volterra competitive system in [7] to systems on time scales. By using comparison theorems of impulsive dynamic equations on time scales, we obtain sufficient conditions to guarantee the permanence of the system. Then based on the Massera-type theorem for impulsive dynamic equations on time scales, we establish existence and uniformly asymptotic stability of the unique positive almost periodic solution of the system. The method of constructing Lyapunov function is skillful. Finally, a difference system with T=1/5ℤ is given to illustrate the feasibility of our results.



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*Source: 2658745-2018-10-09.xml*