# Existence and Stability of Positive Periodic Solutions for  a Neutral Multispecies Logarithmic Population Model with Feedback Control and Impulse

Authors: Zhenguo Luo; Liping Luo
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We investigate a neutral multispecies logarithmic population model with feedback control and impulse. By applying the contraction mapping principle and some inequality techniques, a set of easily applicable criteria for the existence, uniqueness, and global attractivity of positive periodic solution are established. The conditions we obtained are weaker than the previously known ones and can be easily reduced to several special cases. We also give an example to illustrate the applicability of our results.


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## Body

## 1. Introduction

As is known to all, ecosystem in the real world is continuously distributed by unpredictable forces which can result in changes in the biological parameters such as survival rates. Of practical interest in ecology is the question of whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time. In recent years, the qualitative behaviors of the population dynamics with feedback control has attracted the attention of many mathematicians and biologists [1–5]. On the other hand, there are some other perturbations in the real world such as fires and floods, which are not suitable to be considered continually. These perturbations bring sudden changes to the system. Systems with such sudden perturbations involving impulsive differential equations have attracted the interest of many researchers in the past twenty years [6–10], since they provide a natural description of several real processes subject to certain perturbations whose duration is negligible in comparison with the duration of the process. Such processes are often investigated in various fields of science and technology such as physics, population dynamics, ecology, biological systems, and optimal control; for details, see [11–13]. However, to the best of the author’s knowledge, to this day, no scholar considered the neutral multispecies logarithmic population model with feedback control and impulse.The aim of this paper is to investigate the existence, uniqueness, and global attractivity of the positive periodic solution for the following neutral multispecies logarithmic population system with feedback control and impulse:(1)dNi(t)dt=Ni(t)[ri(t)-∑j=1naij(t)lnNj(t)-∑j=1nbij(t)lnNj(t-τij(t))-∑j=1ncij(t)∫-∞tKij(t-s)lnNj(s)ds-∑j=1ndij(t)dlnNj(t-δij(t))dt-ei(t)ui(t)-fi(t)ui(t-σi(t))∑j=1n‍cij(t)∫-∞t‍],t≠tk,dui(t)dt=-αi(t)ui(t)+βi(t)lnNi(t)+ϑi(t)lnNi(t-γi(t)),t≥0,Ni(tk+)=e(1+θik)Ni(tk),i=1,2,…,n,k=1,2,…,
where ui(t) denote indirect feedback control variables. For the ecological justification of (1) and the similar types, refer to [14–20].For the sake of generality and convenience, we always make the following fundamental assumptions:(H1)
ri(t), aij(t), bij(t), cij(t), dij(t), ei(t), fi(t), τij(t), δij(t)∈C2(R,R), σi(t), γi(t), αi(t), βi(t), and ηi(t) are continuous nonnegative ω-periodic functions with ∫0ωri(t)>0, aii(t)>0, δij′(t)<1, τ=maxt∈[0,ω]{τij(t),δij(t),σi(t),γi(t)}, and ∫0∞Kij(s)ds=1, ∫0+∞sKij(s)ds<+∞, i,j=1,2,…,n;(H2)
0<t1<t2<⋯<tk<⋯ are fixed impulsive points with limk→∞tk=+∞;(H3)
{θik} is a real sequence, θik+1>0, and ∏0<tk<t(1+θik) is an ω-periodic function.In the following section, some definitions and some useful lemmas are listed. In the third section, by applying the contraction mapping principle, some sufficient conditions which ensure the existence and uniqueness of positive periodic solution of system (1) are established, and then we get a few sufficient conditions ensuring the global attractivity of the positive periodic solution by employing some inequality techniques. Finally, we give an example to show our results.


## 2. Preliminaries

In order to obtain the existence and uniqueness of a periodic solution for system (1), we first give some definitions and lemmas.Definition 1.
A functionNi:R→(0,∞) (i=1,2,…,n) is said to be a positive solution of (1), if the following conditions are satisfied:(a)
Ni(t) is absolutely continuous on each (tk,tk+1);(b)
for eachk∈Z+, Ni(tk+) and Ni(tk-) exist, and Ni(tk-)=Ni(tk);(c)
Ni(t) satisfies the first equation of (1) for almost everywhere (for short a.e.) in [0,∞]∖{tk} and satisfies Ni(tk+)=(1+θik)Ni(tk) for t=tk, k∈Z+={1,2,…}.Definition 2.
System (1) is said to be globally attractive, if there exists a positive solution (Ni(t),ui(t)) of (1) such that limt→+∞|Ni(t)-Ni*(t)|=0, limt→+∞|ui(t)-ui*(t)|=0, for any other positive solution (Ni*(t),ui*(t)) of the system (1).We can easily get the following lemma.Lemma 3.
R
+
2
n
=
{
(
N
i
(
t
)
,
u
i
(
t
)
)
:
N
i
(
0
)
>
0
,
u
i
(
0
)
>
0
,
i
=
1,2
,
…
,
n
} is the positive invariable region of the system (1).Proof.
In view of biological population, we obtainNi(0)>0, ui(0)>0. By the system (1), we have
(2)Ni(t)=Ni(0)×exp{∫0t[ri(η)-∑j=1naij(η)lnNj(η)-∑j=1nbij(η)lnNj(η-τij(η))-∑j=1ncij(η)∫-∞tKij(η-s)lnNj(s)ds-∑j=1ndij(η)dlnNj(η-δij(η))dt-ei(η)ui(η)-fi(η)ui(η-σi(η))∑j=1ndij(η)dlnNj(η-δij(η))dt]dη},t∈[0,t1],i=1,2,…,n,Ni(t)=Ni(tk)×exp{∫tkt[∑j=1n‍cij(η)∫-∞t‍ri(η)-∑j=1naij(η)lnNj(η)-∑j=1nbij(η)lnNj(η-τij(η))-∑j=1ncij(η)∫-∞tKij(η-s)lnNj(s)ds-∑j=1ndij(η)dlnNj(η-δij(η))dt-ei(η)ui(η)-fi(η)ui(η-σi(η))∑j=1n‍cij(η)∫-∞t‍]dη∫tkt‍},t∈(tk,tk+1],i=1,2,…,n,Ni(tk+)=e(1+pik)Ni(tk)>0,k∈N,i=1,2,…,n,ui(t)=∫tt+ωG(t,s)[(s-γi(s))βi(s)lnNi(s)+ϑi(s)lnNi(s-γi(s))]ds∶=(ϕilnNi)(t),
where
(3)Gi(t,s)=exp{∫tsαi(ξ)dξ}exp{∫tsαi(ξ)dξ}-1.
Then the solution of the system (1) is positive.
Under the above hypotheses(H1)–(H3), we consider the neutral nonimpulsive system:
(4)dyi(t)dt=yi(t)[ri(t)-∑j=1nAij(t)lnyj(t)-∑j=1nBij(t)lnyj(t-τij(t))-∑j=1nCij(t)∫-∞tKij(t-s)lnyj(s)ds-∑j=1nDij(t)dlnyj(t-δij(t))dt-ei(t)ui(t)-fi(t)ui(t-σi(t))∑j=1n‍],dui(t)dt=-αi(t)ui(t)+βi*(t)lnyi(t)+ϑi*(t)lnyi(t-γi(t)),
where
(5)Aij(t)=aij(t)∏0<tk<t(1+θik),Bij(t)=bij(t)∏0<tk<t-τij(t)(1+θik),Cij(t)=cij(t)∏0<tk<t(1+θik),Dij(t)=dij(t)∏0<tk<t-δij(t)(1+θik),βi*(t)=βi(t)∏0<tk<t(1+θik),ϑi*(t)=θi(t)∏0<tk<t-γi(t)(1+θik).
By a solution (yi(t),ui(t)) of (4), it means an absolutely continuous function (yi(t),ui(t)) defined on [-τ,0] that satisfies (4) a.e., for t≥0, and y(ξ)=φ(ξ), y′(ξ)=φ′(ξ) on [-τ,0].The following lemma will be used in the proofs of our results, and the proof of the lemma is similar to that of Theorem 1 in [6].Lemma 4.
Suppose that(H1)–(H4) hold. Then(i)
if(yi(t),ui(t)) is a solution of (4) on [-τ,+∞), then (Ni(t),ui(t))=(∏0<tk<t‍e(1+θik)yi(t),ui(t)) is a solution of (1) on [-τ,+∞),(ii)
if(Ni(t),ui(t)) is a solution of (1) on [-τ,+∞), then (yi(t),ui(t))=(∏0<tk<t‍(e(1+θik))-1Ni(t),ui(t)) is a solution of (4) on [-τ,+∞).Proof.
(i) It is easy to see that(Ni(t),ui(t))=(∏0<tk<t‍e(1+θik)yi(t),ui(t)) is absolutely continuous on every interval (tk,tk+1], t≠tk, k=1,2,…,
(6)Ni′(t)-Ni(t)×[ri(t)-∑j=1n‍aij(t)lnNj(t)-∑j=1n‍bij(t)lnNj(t-τij(t))-∑j=1ncij(t)∫-∞tKij(t-s)lnNj(s)ds-∑j=1n‍dij(t)dlnNj(t-δij(t))dt-ei(t)ui(t)-fi(t)ui(t-σi(t))∑j=1n‍]=∏0<tk<t‍e(1+θik)yi′(t)-∏0<tk<t‍e(1+θik)yi(t)×[ri(t)-∑j=1n‍aij(t)∏0<tk<t‍(1+θik)lnyj(t)-∑j=1n‍bij(t)∏0<tk<t-τij(t)‍(1+θik)lnyj(t-τij(t))-∑j=1n‍cij(t)∫-∞tKj(t-s)∏0<tk<t(1+θik)lnyj(s)ds-∑j=1ndij(t)∏0<tk<t-δij(t)(1+θik)dyj(t-δij(t))dt-ei(t)ui(t)-fi(t)ui(t-σi(t))∑j=1n‍aij(t)∏0<tk<t‍]=∏0<tk<t‍e(1+θk)×{[ri(t)-∑j=1n‍Aij(t)lnyj(t)y′(t)-y(t)×[ri(t)-∑j=1n‍Aij(t)lnyj(t)-∑j=1n‍Bij(t)lnyj(t-τij(t))-∑j=1n‍Cij(t)×∫-∞t‍Kij(t-s)lnyj(s)ds-∑j=1n‍Dij(t)dlnyj(t-δij(t))dt-ei(t)ui(t)-fi(t)ui(t-σi(t))∑j=1n‍]}=0,ui′(t)+αi(t)ui(t)-βi(t)lnNi(t)-ϑi(t)lnNi(t-γi(t))=ui′(t)+αi(t)ui(t)-βi*(t)lnyi(t)-ϑi*(t)lnyi(t-γi(t))=0.
On the other hand, for any t=tk, k=1,2,…,
(7)Ni(tk+)=limt→tk+∏0<tj<t‍e(1+θik)yi(t)=∏0<tj≤tk‍e(1+θik)yi(tk),Ni(tk)=∏0<tj<tk‍e(1+θik)yi(tk).
Thus
(8)N(tk+)=e(1+θik)N(tk),k=1,2,….
It follows from (6)–(8) that (Ni(t),ui(t)) is a solution of (1).
(ii) SinceNi(t)=∏0<tk<t‍e(1+θik)yi(t) is absolutely continuous on every interval (tk,tk+1], t≠tk, k=1,2,…, and in view of (8), it follows that for any k=1,2,…,
(9)yi(tk+)=∏0<tj≤tk‍(e(1+θik))-1Ni(tk+)=∏0<tj<tk‍(e(1+θik))-1Ni(tk)=yi(tk),yi(tk-)=∏0<tj<tk‍(e(1+θik))-1Ni(tk-)=∏0<tj≤tk-‍(e(1+θik))-1Ni(tk-)=yi(tk),
which implies that yi(t) is continuous on [-τ,+∞). It is easy to prove that yi(t) is absolutely continuous on [-τ,+∞). Similar to the proof of (i), we can check that (yi(t),ui(t))=(∏0<tk<t‍(e(1+θik))-1Ni(t),ui(t)) are solutions of (4) on [-τ,+∞). The proof of Lemma 4 is completed.Lemma 5.
(
y
i
(
t
)
,
u
i
(
t
)
) is a ω-periodic solution of (4) if and only if yi(t) is a ω-periodic solution of the following system:
(10)dyi(t)dt=yi(t)[ri(t)-∑j=1n‍Aij(t)lnyj(t)-∑j=1n‍Bij(t)lnyj(t-τij(t))-∑j=1n‍Cij(t)∫-∞t‍Kij(t-s)lnyj(s)ds-∑j=1n‍Dij(t)dlnyj(t-δij(t))dt-ei(t)(ϕilnyi)(t)-fi(t)(ϕilnyi)(t)(t-σi(t))∑j=1n‍],
where
(11)(ϕilnyi)(t)∶=∫tt+ωGi(t,s)[βi*(s)lnyi(s)+ϑi*(s)lnyi(s-γi(s))]ds,
and Gi(t,s) is defined by (3).Proof.
The proof of Lemma5 is similar to that of Lemma 2.2 in [2], and we omit the details here.Obviously, the existence, uniqueness, and global attractivity of positive periodic solution of system (1) is equivalent to the existence, uniqueness, and global attractivity of periodic solution of system (10).Lemma 6.
Assume thatu(t), τ(t) are all continuously differentiable ω-periodic functions and a(t) is a nonnegative continuous ω-periodic function such that ∫0ωa(t)dt>0; then
(12)∫-∞t‍e-∫st‍a(ξ)dξb(s)u′(s-τ(s))ds=c(t)u(t-τ(t))-∫-∞t‍e-∫st‍a(ξ)dξ[a(s)c(s)+c′(s)]u(s-τ(s))ds,
where c(t)=b(t)/(1-τ′(t)).Proof.
As(13)∫-∞t‍e-∫st‍a(ξ)dξb(s)u′(s-τ(s))ds=∫-∞t‍e-∫st‍a(ξ)dξc(s)du(s-τ(s))=e-∫st‍a(ξ)dξc(s)u(s-τ(s))|-∞t-∫-∞t‍u(s-τ(s))d(e-∫st‍a(ξ)dξc(s))=e-∫st‍a(ξ)dξc(s)u(s-τ(s))|-∞t-∫-∞t‍u(s-τ(s))[a(s)c(s)+c′(s)]e-∫st‍a(ξ)dξds.
Denote m=e-∫0ω‍a(t)dt; then from a(t)≥0, ∫0ω‍a(t)dt>0, it follows that m<1. Also, when t≥s without loss of generality, we may assume that s+nω≤t≤s+(n+1)ω; thus
(14)|e-∫st‍a(ξ)dξc(s)u(s-τ(s))|≤e-∫st‍a(ξ)dξ∥c∥∥u∥=e-∑j=1n-1∫s+jωs+(j+1)ω‍a(ξ)dξ-∫s+nωt‍a(ξ)dξ∥c∥∥u∥=mne-∫s+nωt‍a(ξ)dξ∥c∥∥u∥≤mn∥c∥∥u∥.
Therefore,
(15)lims→-∞e-∫st‍a(ξ)dξc(s)u(s-τ(s))=0,
and so from (13)–(15) it follows that
(16)∫-∞t‍e-∫st‍a(ξ)dξb(s)u′(s-τ(s))ds=c(t)u(t-τ(t))-∫-∞t‍e-∫st‍a(ξ)dξ[a(s)c(s)+c′(s)]u(s-τ(s))ds.
The proof Lemma 6 is complete.


## 3. Main Theorem

In this section, by using contraction principle and some inequality techniques, several conditions on the existence, uniqueness, and global attractivity of periodic solution for system (1) are presented.Letyi(t)=ehixi(t); the system (10) can be reduced to
(17)dxidt=-Aii(t)xi(t)-hi-1∑j=1,j≠in‍hjAij(t)xj(t)-hi-1∑j=1n‍hjBij(t)xj(t-τij(t))-hi-1∑j=1n‍hjCij(t)∫-∞t‍Kij(t-s)xj(s)ds-hi-1∑j=1n‍hjDij(t)(1-δij′(t))xj′(t-δij(t))-ei(t)(ϕixi)(t)-fi(t)(ϕixi)(t-σi(t))+hi-1ri(t),
where hi>0  (i=1,2,…,n) are n positive real numbers.Obviously, the existence, uniqueness, and global attractivity of positive periodic solution of system (10) is equivalent to the existence, uniqueness, and global attractivity of periodic solution of system (17).For the rest of this paper, we will devote ourselves to study the existence, uniqueness, and global attractivity of periodic solution of (17). We denote
(18)Γi1(t)∶=ei(t)(ϕi1)(t),Γi2(t)∶=fi(t)(ϕi1)(t-σi(t)),Δi(t)∶=hi-1∑j=1,j≠inhjAij(t)+hi-1∑j=1nhj[(|Dij′(t)|)Bij(t)+Cij(t)+(Aii(s)Dij(t)+|Dij′(t)|)]+Γi1(t)+Γi2(t),i=1,2,…,n.
Our first result on the global existence of a periodic solution of system (1) is stated in the following theorem.Theorem 7.
In addition to(H1)–(H3), assume further that there exist positive constants hi  (i=1,2,…,n) and a positive constant M<1 such that(H4)
supt∈[0,ω]maxi∈[1,n]{∑j=1n(hj/hi)Dij(t)+∫-∞t‍exp{-∫st‍Aii(ξ)dξ}Δi(s)ds}≤M.Then, system (1) has a unique ω-periodic solution with strictly positive components, where Δi(t) is defined by (18).Proof.
From the above analysis, to finish the proof of Theorem7, it is enough to prove under the conditions of Theorem 7 that system (17) has a unique ω-periodic solution. Let
(19)Ω={u(t)∣u∈C(Rn,R),u(t+ω)=u(t)};
under the norm ∥u∥=max1≤i≤nmaxt∈[0,ω]{|ui(t)|}, Ω is a Banach space. For any u(t)=(u1(t),…,un(t))T∈Ω, we consider the periodic solution xu(t) of periodic differential equation
(20)dxidt=-Aii(t)xi(t)-hi-1∑j=1,j≠in‍hjAij(t)uj(t)-hi-1∑j=1n‍hjBij(t)uj(t-τij(t))-hi-1∑j=1n‍hjCij(t)∫-∞t‍Kij(t-s)uj(s)ds-hi-1∑j=1n‍hjDij(t)(1-δij′(t))uj′(t-δij(t))-ei(t)(ϕiui)(t)-fi(t)(ϕiui)(t)(t-σi(t))+hi-1ri(t),i=1,2,…,n.
Since Aii(t)>0, we know that the linear system of system (20)
(21)dxidt=-Aii(t)xi(t),i=1,2,…,n
admits exponential dichotomies on R, and so system (20) has a unique periodic solution xu(t), which can be expressed as
(22)xu(t)=(x1u(t),…,xnu(t))T=(∫-∞t‍exp{-∫st‍A11(ξ)dξ}F1u(s)ds,…,∫-∞t‍exp{-∫st‍Ann(ξ)dξ}Fnu(s)ds)T,
where
(23)Fiu(t)=-hi-1∑j=1,j≠in‍hjAij(t)xj(t)-hi-1∑j=1n‍hjBij(t)xj(t-τij(t))-hi-1∑j=1n‍hjCij(t)∫-∞t‍Kij(t-s)xj(s)ds-hi-1∑j=1n‍hjDij(t)(1-δij′(t))xj′(t-δij(t))-ei(t)(ϕixi)(t)-fi(t)(ϕixi)(t)(t-σi(t))+hi-1ri(t);
its proof is similar to that of Theorem 1 in [18]; here we omit it.
Now, by using Lemma6, xiu(t) can also be expressed as
(24)xiu(t)=-∑j=1n‍hjhiDij(t)uj(t-δij(t))+∫-∞t‍exp{-∫st‍Aii(ξ)dξ}Giu(s)ds,i=1,2,…,n,
where
(25)Giu(t)=-hi-1∑j=1,j≠in‍hjAij(t)uj(t)-hi-1∑j=1n‍hjBij(t)uj(t-τij(t))-hi-1∑j=1n‍hj×Cij(t)∫-∞t‍Kij(t-s)uj(s)ds+hi-1∑j=1n‍hj(Aii(s)Dij(t)+Dij′(t))uj(t-δij(t))+ei(t)(ϕiui)(t)-fi(t)(ϕiui)(t)(t-σi(t))+hi-1ri(t).
Now we define mapping T:Ω→Ω, Tu(t)=xu(t). Following this we will prove that T is a contraction mapping; that is, there exists a constant β∈(0,1), such that ∥Tu-Tv∥≤β∥u-v∥, for all u,v∈Ω. In fact, for any u(t)=(u1(t),…,un(t))T and v(t)=(v1(t),…,vn(t))T, we have
(26)∥Giu(t)-Giv(t)∥≤hi-1∑j=1,j≠in‍hjAij(t)|uj(t)-vj(t)|+hi-1∑j=1n‍hj[∫-∞t‍Bij(t)|uj(t-τij(t))-vj(t-τij(t))|+Cij(t)∫-∞t‍Kij(t-s)|uj(s)-vj(s)|ds+(Aii(s)Dij(t)+Dij′(t))×|uj(t-δij(t))-vj(t-δij(t))|∫-∞t‍]+ei(t)(ϕi1)∥u-v∥+fi(t)(ϕi2)(t-σi(t))∥u-v∥≤{hi-1∑j=1,j≠in‍hjAij(t)+hi-1∑j=1n‍hj[(|Dij′(t)|)Bij(t)+Cij(t)+(Aii(s)Dij(t)+|Dij′(t)|)]+Γi1(t)+Γi2(t)∑j=1,j≠in‍}∥u-v∥=Δi(t)∥u-v∥.
Hence,
(27)∥Tu-Tv∥=supt∈[0,ω]max{|∑j=1nhjh1D1j(t)×[uj(t-δ1j(t))-vj(t-δ1j(t))]∑j=1nhjh1D1j(t)|+∫-∞t‍exp{-∫st‍A11(ξ)dξ}×|G1u(s)-G1v(s)|ds,…,|∑j=1n‍hjhnDnj(t)×[uj(t-δnj(t))-vj(t-δnj(t))]∑j=1n‍|+∫-∞t‍exp{-∫st‍Ann(ξ)dξ}×|Gnu(s)-Gnv(s)|ds|∑j=1nhjh1D1j(t)∫st‍}≤supt∈[0,ω]max{[∑j=1n‍hjh1D1j(t)+∫-∞t‍exp{-∫st‍A11(ξ)dξ}Δ1(s)ds∑j=1n‍hjh1]×∥u-v∥,…,[∑j=1n‍hjhnDnj(t)+∫-∞t‍exp{-∫st‍Ann(ξ)dξ}Δn(s)ds∑j=1n‍hjh1]×∥u-v∥∑j=1n‍hjh1∑j=1n‍}.
It follows from (H4) that ∥Tu-Tv∥≤∥u-v∥ for all u,v∈Ω. That is, T is a contraction mapping. Hence, there exists a unique fixed point x*(t)=(x1*(t),…,xn*(t))T∈Ω; that is, Tx*(t)=x*(t). Therefore, x*(t) is the unique periodic solution of system (17). It follows from (1), (4), (10), and (17) that (N*(t),u*(t))T=(N1*(t),…,Nn*(t),u1*(t),…,un*(t))T is the unique positive periodic solution of system (1). The proof of Theorem 7 is completed.Our next theorem is concerned with the global stability of periodic solution for system (1).Theorem 8.
In addition to(H1)–(H4), suppose further that the following condition holds:(H5)
exp{-∫0t‍Aii(ξ)dξ}→0, as t→+∞, i=1,2,…,n.Then system (1) has a unique periodic solution which is globally attractive.Proof.
LetN*(t)=(N1*(t),N2*(t),…,Nn*(t))T be the unique positive periodic solution of system (1), whose existence and uniqueness are guaranteed by Theorem 7, and let N(t)=(N1(t),N2(t),…,Nn(t))T be any other solution of system (1). Let Ni*(t)=exp{∏0<tk<t‍hi(1+pik)xi*(t)}, Ni(t)=exp{∏0<tk<t‍×hi(1+pik)xi(t)}; then, similar to (17), we have
(28)dxi*dt=-Aii(t)xi*(t)-hi-1∑j=1,j≠in‍hjAij(t)xj*(t)-hi-1∑j=1n‍hjBij(t)xj*(t-τij(t))-hi-1∑j=1n‍hjCij(t)∫-∞t‍Kij(t-s)xj*(s)ds-hi-1∑j=1n‍hjDij(t)(1-δij′(t))xj′*(t-δij(t))-ei(t)(ϕixi*)(t)-fi(t)(ϕixi*)(t-σi(t))+hi-1ri(t),dxidt=-Aii(t)xi(t)-hi-1∑j=1,j≠in‍hjAij(t)xj(t)-hi-1∑j=1n‍hjBij(t)xj(t-τij(t))-hi-1∑j=1n‍hjCij(t)∫-∞t‍Kij(t-s)xj(s)ds-hi-1∑j=1n‍hjDij(t)(1-δij′(t))xj′(t-δij(t))-ei(t)(ϕixi)(t)-fi(t)(ϕixi)(t-σi(t))+hi-1ri(t).
Let xi*(t)-xi(t)=wi(t); then
(29)dwidt=-Aii(t)wi(t)-hi-1∑j=1,j≠in‍hjAij(t)wj(t)-hi-1∑j=1n‍hjBij(t)wj(t-τij(t))-hi-1∑j=1n‍hjCij(t)∫-∞t‍Kij(t-s)wj(s)ds-hi-1∑j=1n‍hjDij(t)(1-δij′(t))wj′(t-δij(t))-ei(t)(ϕiwi)(t)-fi(t)(ϕiwi)(t-σi(t)).
Multiply both sides of (29) with exp{∫0t‍Aii(ξ)dξ}, and then integrate from 0 to t to obtain
(30)∫0t‍[wi(u)exp{∫0u‍Aii(ξ)dξ}]′du=-∫0t‍[hi-1∑j=1,j≠in‍hjAij(u)wj(u)+hi-1∑j=1n‍hjBij(u)wj(u-τij(u))+hi-1∑j=1n‍hjCij(u)∫-∞t‍Kij(u-s)wj(s)ds+hi-1∑j=1n‍hjDij(u)(1-δij′(u))wj′(u-δij(u))+ei(u)(ϕiwi)(u)+fi(u)(ϕiwi)(u-σi(t))∑j=1n‍]×exp{∫0u‍Aii(ξ)dξ}du,i=1,2,…,n;
then
(31)wi(t)=wi(0)exp{-∫0t‍Aii(ξ)dξ}-∫0t‍[hi-1∑j=1,j≠in‍hjAij(u)wj(u)+hi-1∑j=1n‍hjBij(u)wj(u-τij(u))+hi-1∑j=1n‍hjCij(u)×∫-∞t‍Kij(u-s)wj(s)ds+hi-1∑j=1n‍hjDij(u)(1-δij′(u))×wj′(u-δij(u))+ei(u)(ϕiwi)(u)+fi(u)(ϕiwi)(u-σi(t))∑j=1,j≠in‍]×exp{-∫ut‍Aii(ξ)dξ}du,i=1,2,…,n.
Let D0ij(t)=Dij(t)(1-δij′(t)); we see that
(32)∫0t‍D0ij(u)wj′(u-δij(u))exp{-∫ut‍Aii(ξ)dξ}du=∫0t‍D0ij(u)wj′(u-δij(u))(1-δij′(u))1-δij(u)×exp{-∫ut‍Aii(ξ)dξ}du=∫0t‍[D0ij(u)exp{-∫ut‍Aii(ξ)dξ}1-δij′(u)]×[wj′(u-δij(u))(1-δij′(u))]du=[D0ij(t)1-δij′(t)wj(t-δij(t))-D0ij(0)1-δij′(0)wj(-δij(0))exp{-∫0tAii(ξ)dξ}]-∫0t‍(Aii(u)Dij(u)+Dij′(u))×exp{-∫ut‍Aii(ξ)dξ}wj(u-δij(u))du=[∫0t‍Dij(t)wj(t-δij(t))-Dij(0)wj(-δij(0))exp{-∫0t‍Aii(ξ)dξ}]-∫0t‍(Aii(u)Dij(u)+Dij′(u))×exp{-∫ut‍Aii(ξ)dξ}wj(u-δij(u))du.
Substituting (32) into (31), we get
(33)wi(t)=[wi(0)+hi-1∑j=1n‍hjDij(0)wj(-δij(0))]×exp{-∫0t‍Aii(ξ)dξ}+∫0t‍{hi-1∑j=1n‍hj[∫-∞t‍(Aii(u)Dij(u)+Dij′(u))wj(u-δij(u))-Bij(u)wj(u-τij(u))-Cij(u)∫-∞t‍Kij(u-s)wj(s)ds]-hi-1∑j=1,j≠in‍hjAij(u)wj(u)-ei(u)(ϕiwi)(u)-fi(u)(ϕiwi)(u-σi(t))∑j=1n‍}×exp{-∫ut‍Aii(ξ)dξ}du+hi-1∑j=1n‍hjDij(t)wj(t-δij(t));
therefore, we have
(34)∥w∥≤|wi(0)+hi-1∑j=1n‍hjDij(0)wj(-δij(0))|×exp{-∫0t‍Aii(ξ)dξ}+{∫0t‍{hi-1∑j=1n‍hj[(Aii(u)Dij(u)+|Dij′(u)|)×|Bij(u)|+|Cij(u)|]+hi-1∑j=1,j≠in‍hj|Aij(u)|+|Γi1(u)|+|Γi2(u)|}×exp{-∫ut‍Aii(ξ)dξ}du+hi-1∑j=1n‍hj|Dij(t)|}∥w∥=|wi(0)+hi-1∑j=1n‍hjDij(0)wj(-δij(0))|×exp{-∫ut‍Aii(ξ)dξ}+[∑j=1n‍hjhiDij(t)+∫0t‍exp{-∫ut‍Aii(ξ)dξ}Δi(u)du∑j=1n‍]∥w∥,
where Δi(t) is defined by (18). From (H4), we have
(35)∥w∥≤|wi(0)+hi-1∑j=1nhjDij(0)wj(-δij(0))|exp{-∫0tAii(ξ)dξ}1-hi-1∑j=1nhj|Dij(t)|-∫0tΔi(u)exp{-∫ut‍Aii(ξ)dξ}du≤|wi(0)+hi-1∑j=1nhjDij(0)wj(-δij(0))|exp{-∫0tAii(ξ)dξ}1-M.
From (H5), we have
(36)∥w∥=maxi∈[0,n]|wi(t)|=maxi∈[0,n]|xi*(t)-xi(t)|=0,ast⟶+∞;
thus, xi(t)→xi*(t), as t→+∞, i=1,2,…,n. Hence, the positive ω-periodic solution of (17) is globally attractive; accordingly, Ni(t)=exp{∏0<tk<t‍hi(1+pik)xi(t)}→exp{∏0<tk<t‍hi(1+pik)xi*(t)}=Ni*(t), ui(t)=(ϕilnNi)(t)→(ϕilnNi*)(t)=ui*(t) as t→+∞, i=1,2,…,n, and by Definition 2, the positive ω-periodic solution of (1) is globally attractive. The proof of Theorem 8 is completed.Remark 9.
Ifei(t)=fi(t)=αi(t)=βi(t)=ϑi(t)=0, θik+1=0, i=1,2,…,n, k=1,2,…, then system (1) is studied by [3]. Hence, Theorems 7 and 8 generalize the corresponding results in [3].Remark 10.
Ifθik+1=0, i=1,2,…,n, k=1,2,…, then system (1) is studied by [4]. Hence Theorems 7 and 8 also generalize the corresponding results in [4].


## 4. Example

Consider the following impulsive model:(37)dN(t)dt=N(t)[∫-∞t‍r(t)-a(t)lnN(t)-b(t)lnN(t-τ(t))-c(t)∫-∞t‍K(t-s)lnN(s)ds-d(t)dlnN(t-δ(t))dt-e(t)u(t)-f(t)u(t-σ(t))∫-∞t‍],t≠tk,du(t)dt=-α(t)u(t)+β(t)lnN(t)+θ(t)lnN(t-γ(t)),t≥0,N(tk+)=e(1+pk)N(tk),k=1,2,…,
where r(t), a(t), b(t), c(t), d(t), e(t), f(t), δ(t)∈C2(R,R), σ(t), γ(t), α(t), β(t), θ(t) are all nonnegative ω-periodic continuous functions with ∫0ωr(t)>0, a(t)>0, δ′(t)<1 and pk is a real sequence, and ∏0<tk<t(1+pk) is a positive ω-periodic function with k=1,2,…. Furthermore, ∫0∞K(s)ds=1, ∫0+∞sK(s)ds<+∞.We denote(38)A(t)=a(t)∏0<tk<t(1+pk),B(t)=b(t)∏0<tk<t-τ(t)(1+pk),C(t)=c(t)∏0<tk<t(1+pk),D(t)=d(t)∏0<tk<t-δ(t))(1+pk),D0(t)=D(t)(1-δ′(t)),β*(t)=β(t)∏0<tk<t(1+pk)θ*(t)=θ(t)∏0<tk<t-γ(t)(1+pk),Γ1(t)∶=e(t)(ϕ1)(t),Γ2(t)∶=f(t)(ϕ1)(t-σ(t)),Δ(t)∶=B(t)+C(t)+A(t)D(t)+|D′(t)|+Γ1(t)+Γ2(t).
Similar to Theorems 7 and 8, we can get the following results.Corollary 11.
In addition to conditions(H1)–(H3), assume further that there exists a positive constant M<1 such that(H6)
D(t)+∫-∞texp{-∫stA(ξ)dξ}Δ(s)ds≤M.Then, (37) has a unique ω-periodic solution with strictly positive components, where Δ(t) is defined by (38).Corollary 12.
In addition to conditions(H1)–(H3) and (H6), suppose further that the following condition holds:(H7)
exp{-∫0tA(ξ)dξ}→0, as t→+∞.Then, system (37) has a unique periodic solution which is globally attractive.Remark 13.
The results in the work show that by means of appropriate impulsive perturbations and feedback control we can control the dynamics of these equations.



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*Source: 741043-2013-10-02.xml*