# Global Attractivity of a Periodic DelayedN-Species Model of Facultative Mutualism

Authors: Ahmadjan Muhammadhaji; Zhidong Teng
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

Two classes of periodicN-species Lotka-Volterra facultative mutualism systems with distributed delays are discussed. Based on the continuation theorem of the coincidence degree theory developed by Gaines and Mawhin and the Lyapunov function method, some new sufficient conditions on the existence and global attractivity of positive periodic solutions are established.


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## Body

## 1.   Introduction

Mutualism is the interaction of two species of organisms that benefits both [1]. In general, mutualism may be either obligate or facultative. Obligate mutualist may survive only by association, and facultative mutualist, while benefiting from the presence of each other, may also survive in the absence of any of them [2]. As it is well known, in recent years the nonautonomous and periodic population dynamical systems are extensively studied. The basic and important studied questions for these systems are the persistence, permanence, and extinction of species, global stability of systems and the existence of positive periodic solutions, positive almost periodic solutions and strictly positive solutions, and so forth. Many important and influential results have been established and can be found in many articles and books. Particularly, the existence of positive periodic solutions for various type population dynamical systems has been extensively studied in [1–16] and the references cited therein.In [7], the authors studied the following delayed two-species model of facultative mutualism:
(1)y˙1(t)=y1(t)[r1(t)-a1(t)y1(t)-b1(t)y1(t-τ1(t))y1(t)+c1(t)y2(t-p1(t))],y˙2(t)=y2(t)[r2(t)-a2(t)y2(t)-b2(t)y2(t-τ1(t))y2(t)+c2(t)y1(t-p1(t))].
By using the technique of coincidence degree and the Lyapunov functionals method, the sufficient conditions for the existence and globally asymptotic stability of positive periodic solutions are obtained for system (1). In [2], the authors considered the following periodic delayed two-species model of facultative mutualism:
(2)y˙1(t)=y1(t)[r1(t)-a1(t)y1(t)+b1(t)y1(t-τ1(t))y1(t)+c1(t)y2(t-p1(t))],y˙2(t)=y2(t)[r2(t)-a2(t)y2(t)+b2(t)y2(t-τ1(t))y2(t)+c2(t)y1(t-p1(t))].
By means of the methods of coincidence degree and the Lyapunov functional, the sufficient conditions for the existence and globally asymptotic stability of positive periodic solutions are established for system (2). In [12], the following n-species periodic Lotka-Volterra type competitive systems with feedback controls and finite and infinite distributed delays are discussed:
(3)y˙i(t)=yi(t)[ri(t)-aii(t)yi(t)∫0ω‍yi(t)t-∑j=1,j≠in‍aij(t)∫0ω‍Kij(s)yj(t-s)dsyi(t)t-αi(t)∫0ω‍Hi(s)ui(t-s)ds],u˙i(t)=-ηi(t)+ai(t)∫0ω‍Ki(s)yi(t-s)ds,
where i=1,2,…,n. By using the technique of coincidence degree and the Lyapunov functionals method, the sufficient conditions for the existence and global stability of positive periodic solutions are obtained for system (3).Motivated by the above works, in this paper, we investigate the following two classes ofn species periodic model of facultative mutualism with finite distributed delays:
(4)x˙i(t)=xi(t)[ri(t)-αi(t)xi(t)∑j≠in‍∫-τ0‍xi(t)+∑j=1n‍∑l=1m‍aijl(t)∫-τ0‍kijl(s)xj(t+s)ds]i=1,2,…n,(5)x˙i(t)=xi(t)[ri(t)-αi(t)xi(t)∑j≠in‍∫-τ0‍xi(t)-∑l=1m‍aiil(t)∫-τ0‍kiil(s)xi(t+s)dsxi(t)+∑j≠in‍∑l=1m‍aijl(t)∫-τ0‍kijl(s)xj(t+s)ds]i=1,2,…n.By using the technique of coincidence degree developed by Gaines and Mawhin in [17] and the Lyapunov functional method, we will establish some new sufficient conditions which guarantee that the system has at least one positive periodic solution and is globally attractive.The organization of this paper is as follows. In the next section we will present some basic assumptions and main definitions and lemmas. In Section3, conditions for the existence and global attractivity of positive periodic solution. In Section 4, two examples are given to illustrate that our main results are applicable. In the final section, we will discuss what we study in this paper and what we had in this paper.


## 2. Preliminaries

In systems (4) and (5), we have that xi(t)(i=1,2,…n) represent the density of n species xi(i=1,2,…n) at time t, respectively; ri(t)(i=1,2,…n) represent the intrinsic growth rate of species xi(i=1,2,…n) at time t, respectively; αi(t)(i=1,2,…,n) represent the intrapatch restriction density of species xi(i=1,2,…n) at time t, respectively; aijl(t)(l=1,2,…,m,i≠j,i,j=1,2,…,n) represent the mutualism coefficients between n species xi(i=1,2,…n) at time t, respectively, while τ≥0 is a constant and τ may be +∞. System (4) involves positive feedback terms
(6)∑l=1m‍aiil(t)∫-τ0‍kiil(s)xi(t+s)ds,i=1,2,…,n
which can be regarded as the passive effect on the growth rate of a species, and system (5) involves negative feedback terms
(7)-∑l=1m‍aiil(t)∫-τ0‍kiil(s)xi(t+s)ds,i=1,2,…,n
which are due to gestation. In this paper, we always assume the following:(H1)
ri(t) are ω-periodic continuous functions with ∫0ω‍ri(t)dt>0(i=1,2,…,n); αi(t)(i=1,2,…,n) and aijl(t)(i,j=1,2,…,n;l=1,2,…,m) are positive ω-periodic continuous functions; kijl(s)(i,j=1,2,…,n;l=1,2,…,m) are nonnegative integrable functions on [-τ,0] satisfying ∫-τ0‍kijl(s)ds=1.From the viewpoint of mathematical biology, in this paper for systems (4) and (5) we only consider the solution with the following initial condition:
(8)xi(t)=ϕi(s),∀s∈[-τ,0],i=1,2,…,n,
where ϕi(s)(i=1,2,…,n) are nonnegative continuous functions defined on [-τ,0] satisfying ϕi(0)>0(i=1,2,…,n).In this paper, for anyω-periodic continuous function f(t) we denote the following:
(9)fL=mint∈[0,ω]f(t),fM=maxt∈[0,ω]f(t),f-=1ω∫0ω‍f(t)dt.In order to obtain the existence of positiveω-periodic solutions of systems (4) and (5), we will use the continuation theorem developed by Gaines and Mawhin in [17]. For the reader’s convenience, we will introduce the continuation theorem in the following.LetX and Z be two normed vetor spaces. Let L:DomL⊂X→Z  be a linear operator and N:X→Z  be a continuous operator. The operator L is called a Fredholm operator of index zero, if  dimKerL=codimImL<∞ and ImL is a closed set in Z. If L is a Fredholm operator of index zero, then there exist continuous projectors P:X→X and Q:Z→Z such that ImP=KerL and ImL=KerQ=Im(I-Q). It follows that L∣DomL∩KerP:DomL∩KerP→ImL is invertible and its inverse is denoted by KP and denote by J:ImQ→KerL an isomorphism of ImQ onto Ker L. Let Ω be a bounded open subset of X, we say that the operator N is L-compact on Ω¯, where Ω¯ denotes the closure of Ω in X, if QN(Ω¯) is bounded and KP(I-Q)N:Ω¯→X is compact.Lemma 1 (see [17]).
LetL be a Fredholm operator of index zero and let N be L-compact on Ω¯. If(a)
for eachλ∈(0,1) and x∈∂Ω∩DomL, Lx≠λNx;(b)
for eachx∈∂Ω∩KerL, QNx≠0;(c)
deg{JQN,Ω∩KerL,0}≠0,then the operator equation Lx=Nx has at least one solution lying in DomL∩Ω¯.


## 3. Main Results

Now, for the convenience of statements, we denote the function(10)aij(t)=∑l=1m‍aijl(t),i,j=1,2,…,n.The following theorem is about the existence and global attractivity of positive periodic solutions of system (4).Theorem 2.
Suppose that assumption(H1) holds and there exists a constant μi>0(i=1,2,…,n) such that
(11)mint∈[0,ω]{μiαi(t)-∑j=1n‍∑l=1m‍μj∫-τ0‍ajil(t-s)kjil(s)ds}=:δi>0,i=1,2,…,n,
and the algebraic equation
(12)r-i-α-ivi+∑j=1n‍a-ijvj=0,i=1,2,…,n
has a unique positive solution. Then, system (4) has a positive ω-periodic solution which is globally attractive.Proof.
We firstly consider the existence of positive periodic solutions of system (4). For system (4), we introduce new variables ui(t)(i=1,2,…,n) such that
(13)xi(t)=exp{ui(t)},i=1,2,…,n.
Then, system (4) is rewritten in the following form:
(14)u˙i(t)=ri(t)-αi(t)exp{ui(t)}+∑l=1m‍aiil(t)∫-τ0‍kiil(s)exp{ui(t+s)}ds+∑j≠in‍∑l=1m‍aijl(t)∫-τ0‍kijl(s)exp{uj(t+s)}ds,i=1,2,…,n.
In order to apply Lemma 1 to system (14), we introduce the normed vector spaces X and Z as follows. Let C(R,Rn) denote the space of all continuous function u(t)=(u1(t),u2(t),…,un(t)):R→Rn. We take
(15)X=Z={u(t)∈C(R,Rn):u(t)zisanω-periodicfunction}
with norm
(16)∥u∥=∑i=1n‍maxt∈[0,ω]|ui(t)|.
It is obvious that X and Z are the Banach spaces.
We define a linear operatorL:DomL⊂X→Z and a continuous operator N:X→Z as follows:
(17)Lu(t)=u˙(t),Nu(t)=(Nu1(t),Nu2(t),…,Nun(t)),
where
(18)Nui(t)=ri(t)-αi(t)exp{ui(t)}+∑l=1m‍aiil(t)∫-τ0‍kiil(s)exp{ui(t+s)}ds+∑j≠in‍∑l=1m‍aijl(t)∫-τ0‍kijl(s)exp{uj(t+s)}ds,i=1,2,…,n.
Further, we define continuous projectors P:X→X and Q:Z→Z as follows:
(19)Pu(t)=1ω∫0ω‍u(t)dt,Qv(t)=1ω∫0ω‍v(t)dt.
We easily see that  ImL={v∈Z:∫0ω‍v(t)dt=0} and KerL=Rn. It is obvious that ImL is closed in Z and dimKerL=n. Since for any v∈Z there are unique v1∈Rn and v2∈ImL with
(20)v1=1ω∫0ω‍v(t)dt,v2(t)=v(t)-v1,
such that v(t)=v1+v2(t), we have co dim ImL=n. Therefore, L is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to L) Kp:ImL→KerP∩DomL is given in the following form:
(21)Kpv(t)=∫0t‍v(s)ds-1ω∫0ω‍∫0t‍v(s)dsdt.
For convenience, we denote F(t)=(F1(t),F2(t),…,Fn(t)) as follows:
(22)Fi(t)=ri(t)-αi(t)exp{ui(t)}+∑l=1m‍aiil(t)∫-τ0‍kiil(s)exp{ui(t+s)}ds+∑j≠in‍∑l=1m‍aijl(t)∫-τ0‍kijl(s)exp{uj(t+s)}ds,i=1,2,…n.
Thus, we have
(23)QNu(t)=1ω∫0ω‍F(t)dt,Kp(I-Q)Nu(t)=KpINu(t)-KpQNu(t)=∫0t‍F(s)ds-1ω∫0ω‍∫0t‍F(s)dsdt+(12-tω)∫0ω‍F(s)ds.
From formulas (23), we easily see that QN and Kp(I-Q)N are continuous operators. Furthermore, it can be verified that Kp(I-Q)N(Ω¯)¯ is compact for any open bounded set Ω⊂X by using Arzela-Ascoli theorem and QN(Ω¯) is bounded. Therefore, N is L-compact on Ω¯ for any open bounded subset Ω⊂X.
Now, we reach the position to search for an appropriate open bounded subsetΩ for the application of the continuation theorem (Lemma 1) to system (4).
Corresponding to the operator equationLu(t)=λNu(t) with parameter λ∈(0,1), we have
(24)u˙i(t)=λFi(t),i=1,2,…,n,
where Fi(t)(i=1,2,…,n) is given in (22).
Assume thatu(t)=(u1(t),u2(t),…,un(t))∈X is a solution of system (24) for some parameter λ∈(0,1). By integrating system (24) with the interval [0,ω], we obtain the following:
(25)∫0ω‍[ri(t)-αi(t)exp{ui(t)}∑j≠in∫-τ0‍‍+∑l=1m‍aiil(t)∫-τ0‍kiil(s)exp{ui(t+s)}ds+∑j≠in‍∑l=1m‍aijl(t)∫-τ0‍kijl(s)exp{uj(t+s)}ds]dt=0,i=1,2,…n.
Consequently,
(26)∫0ω‍[αi(t)exp{ui(t)}∑j≠in‍∫-τ0‍-∑l=1m‍aiil(t)∫-τ0‍kiil(s)exp{ui(t+s)}ds-∑j≠in‍∑l=1m‍aijl(t)∫-τ0‍kijl(s)exp{uj(t+s)}ds]dt=r-iω,i=1,2,…n.
From the continuity of u(t)=(u1(t),u2(t),…,un(t)), there exist constants ξi,ηi∈[0,ω](i=1,2,…,n) such that
(27)ui(ξi)=maxt∈[0,ω]ui(t),ui(ηi)=mint∈[0,ω]ui(t),i=1,2,…,n.
From (26) and (27), we obtain
(28)∫0ω‍αi(t)exp{ui(ξi)}dt≥r-iω,i=1,2,…,n.
Therefore, we further have
(29)ui(ξi)≥ln(r-iα-i),i=1,2,…,n.
For each i,j=1,2,…,n and l=1,2,…,m, we have
(30)∫0ω‍aijl(t)∫-τ0‍kijl(s)exp{uj(t+s)}dsdt=∫-τ0‍∫0ω‍aijl(t)kijl(s)exp{uj(t+s)}dtds=∫-τ0‍∫ss+ω‍aijl(v-s)kijl(s)exp{uj(v)}dvds=∫-τ0‍∫0ω‍aijl(v-s)kijl(s)exp{uj(v)}dvds=∫0ω‍∫-τ0‍aijl(v-s)kijl(s)exp{uj(v)}dsdv=∫0ω‍(∫-τ0‍aijl(t-s)kijl(s)ds)exp{uj(t)}dt.
Hence, from (26) we further obtain
(31)∫0ω‍[(αi(t)-∑l=1m‍(∫-τ0‍aiil(t-s)kiil(s)ds))exp{ui(t)}∑j≠in‍-∑j≠in‍∑l=1m‍(∫-τ0‍aijl(t-s)kijl(s)ds)exp{uj(t)}]dt=r-iω,i=1,2,…,n.
Consequently,
(32)∫0ω[∑j≠1n‍∑l=1m‍(∫-τ0‍a1jl(t-s)k1jl(s)ds)(α1(t)-∑l=1m‍(∫-τ0‍a11l(t-s)k11l(s)ds))exp{u1(t)}‍-∑j≠1n‍∑l=1m‍(∫-τ0‍a1jl(t-s)k1jl(s)ds)exp{uj(t)}]dt+∫0ω[(α2(t)-∑l=1m‍(∫-τ0‍a22l(t-s)k22l(s)ds))‍×exp{u2(t)}-∑j≠2n‍∑l=1m‍(∫-τ0‍a2jl(t-s)k2jl(s)ds)×exp{uj(t)}(∑l=1m‍(∫-τ0‍a22l(t-s)))]dt+⋯+∫0ω[(αn(t)∫-τ0‍‍-∑l=1m‍(∫-τ0‍annl(t-s)knnl(s))ds∑l=1m‍)×exp{un(t)}-∑j≠nn‍∑l=1m‍(∫-τ0‍anjl(t-s)knjl(s)ds)×exp{uj(t)}(αn(t)∫-τ0‍]dt=∫0ω[α1(t)-∑l=1m(∫-τ0‍a11l(t-s)k11l(s)ds∑j≠1n‍‍‍+∑j≠1n‍∫-τ0‍aj1l(t-s)kj1l(s)ds)]×exp{u1(t)}dt+∫0ω‍[α2(t)-∑l=1m(∫-τ0‍a22l(t-s)k22l(s)ds‍+∑j≠2n‍∫-τ0‍aj2l(t-s)×kj2l(s)ds∫-τ0‍)∑l=1m‍]×exp{u2(t)}dt+⋯+∫0ω[αn(t)×kjnl(s)ds∫-τ0‍)∑l=1m(∫-τ0‍annl(t-s)knnl(s)ds‍‍-∑l=1m(∫-τ0‍annl(t-s)knnl(s)ds‍+∑j≠nn‍∫-τ0‍ajnl(t-s)×kjnl(s)ds∫-τ0‍)∑l=1m(∫-τ0‍annl(t-s)knnl(s)ds‍]×exp{un(t)}dt=∫0ω[×kj1l(s)∑j≠1n‍](∫-τ0‍ds)α1(t)‍-∑l=1m‍(∫-τ0‍[a11l(t-s)k11l(s)+∑j≠1n‍aj1l(t-s)×kj1l(s)∑j≠1n‍]ds∫-τ0‍)]×exp{u1(t)}dt+∫0ω‍[α2(t)-∑l=1m‍(∫-τ0‍[a22l(t-s)k22l(s)kj2l(∫-τ0‍)+∑j≠2n‍aj2l(t-s)×kj2l(s)∫-τ0‍]ds∫-τ0‍)∑l=1m‍]×exp{u2(t)}dt+⋯+∫0ω[∫-τ0‍∑j≠nn‍αn(t)×kjnl(s)]∫-τ0‍)ds]‍-∑l=1m‍(∫-τ0‍[∑j≠nn‍annl(t-s)knnl(s)∑j≠nn‍kjnl+∑j≠nn‍ajnl(t-s)×kjnl(s)∑j≠nn‍]ds∫-τ0‍)]×exp{un(t)}dt=∑i=1n‍r-iω.
From the assumptions of Theorem 2, we can obtain
(33)∫0ω[αi(t)-∑l=1m‍(∫-τ0[aiil(t-s)kiil(s)kjil‍+∑j≠in‍ajil(t-s)kjil(s)]‍+∑j≠in‍ajil(t-s)×kjil(s)]ds∫-τ0‍aiil(t-s)kiil(s)kjil)+∑j≠in‍ajil(t-s)kjil(s)]×exp{ui(t)}dt≤∑i=1n‍r-iω,i=1,2,…,n.
Hence,
(34)δi∫0ω‍exp{ui(t)}dt≤∑i=1n‍r-iω,i=1,2,…,n.
Consequently,
(35)∫0ω‍exp{ui(t)}dt≤∑i=1nr-iωδi,i=1,2,…,n.
From (35), we further obtain
(36)ui(ηi)≤ln(∑i=1nr-iδi),i=1,2,…,n.
On the other hand, directly from system (14) we have
(37)∫0ω‍|u˙i(t)|dt≤∫0ω‍[|ri(t)|+αi(t)exp{ui(t)}∑j≠1n‍∫-τ0‍+∑l=1m‍aiil(t)∫-τ0‍kiil(s)exp{ui(t+s)}ds+∑j≠1n‍∑l=1m‍aijl(t)∫-τ0‍kijl(s)×exp{uj(t+s)}ds∑j≠1n∫-τ0‍‍]dt=∫0ω‍|ri(t)|dt+∫0ω‍[αi(t)+∑l=1m‍∫-τ0‍aiil(t-s)kiil(s)ds]×exp{ui(t)}dt+∫0ω‍(∑j≠in‍∑l=1m‍∫-τ0‍aijl(t-s)kijl(s)ds)×exp{uj(t)}dt≤∫0ω‍|ri(t)|dt+∫0ω‍αi(t)exp{ui(t)}dt+∑l=1m‍aiilM∫0ω‍exp{ui(t)}dt+∑j≠in‍∑l=1m‍aijlM∫0ω‍exp{uj(t)}dt≤|ri|-ω+∑j=1n‍∑l=1m‍aijlM∑i=1nr-iωδi+αiM∑i=1nr-iωδi,i=1,2,…,n.
From (36) and (37), we have, for any t∈[0,ω],
(38)ui(t)≤ui(ηi)+∫0ω‍|u˙i(t)|dt≤ln(∑i=1nr-iδi)+|ri|-ω+∑j=1n‍∑l=1m‍aijlM∑i=1nr-iωδi+αiM∑i=1nr-iωδi=:Mi,i=1,2,…,n.
Further, from (29) and (37), we have, for any t∈[0,ω],
(39)ui(t)≥ui(ξi)-∫0ω‍|u˙i(t)|dt≥ln(r-iα-i)-|ri|-ω-∑j=1n‍∑l=1m‍aijlM∑i=1nr-iωδi-αiM∑i=1nr-iωδi=:Ni,i=1,2,…,n.
Therefore, from (38) and (39) we have
(40)maxt∈[0,ω]|ui(t)|≤max{|Mi|,|Ni|}=:Bi,i=1,2,…,n.
It can be seen that the constants Bi(i=1,2,…,n) are independent of parameter λ∈(0,1).
For anyu=(u1,u2,…,un)∈Rn, from (18) we can obtain
(41)QNu=(QNu1,QNu2,…,QNun),
where
(42)QNu=r-i-(α-i-a-ii)exp{ui}+∑j≠in‍a-ijexp{uj},i=1,2,…,n.
We consider the following algebraic equation:
(43)r-i-(α-i-a-ii)vi+∑j≠in‍a-ijvj=0,i=1,2,…,n.
From the assumption of Theorem 2, the equation has a unique positive solution v*=(v1*,v2*,…,vn*). Hence, the equation QNu=0 has a unique solution u*=(u1*,u2*,…,un*)=(lnv1*,lnv2*,…,lnvn*)∈Rn.
Choosing constantB>0 large enough such that |u1*|+|u2*|+⋯+|un*|<B and B>B1+B2+⋯+Bn, we define a bounded open set Ω⊂X as follows:
(44)Ω={u∈X:∥u∥<B}.
It is clear that Ω satisfies conditions (a) and (b) of Lemma 1. On the other hand, by direct calculating we can obtain
(45)deg{JQN,Ω∩KerL,(0,0,…,0)}=sgn|fu11fu21⋯fun1fu12fu22⋯fun2⋯⋯⋯⋯fu1nfu2n⋯funn|,
where
(46)fuji=-(α-i-a-ij)exp{uj*},i=j,fuji=a-ijexp{uj*},i≠j,i,j=1,2,…,n.
From the assumption of Theorem2, we have
(47)|fu11fu21⋯fun1fu12fu22⋯fun2⋯⋯⋯⋯fu1nfu2n⋯funn|≠0.
From this, we finally have(48)deg{JQN,Ω∩KerL,(0,0,…,0)}≠0.
This shows that Ω satisfies condition (c) of Lemma 1. Therefore, system (14) has a ω-periodic solution u*(t)=(u1*(t),u2*(t),…,un*(t))∈Ω-. Further, from (13), system (4) has a positive ω-periodic solution x*(t)=(x1*(t),x2*(t),…,xn*(t)).
Next, we will consider the global attractivity of positive periodic solutionsx*(t)=(x1*(t),x2*(t),…,xn*(t)) of system (4). Choose positive constants mi>0, Mi>0 such that
(49)mi≤xi*(t)≤Mi,i=1,2,…,n.
From the assumption of Theorem 2, there exists constant β>0 such that for all t≥0 we have
(50)δi≥β>0,i=1,2,…,n.
Let (x1(t),x2(t),…,xn(t)) be any solution of system (4), we define Lyapunov function as follows:
(51)Vi(t)=μi|lnxi*(t)-lnxi(t)|+∑j=1n‍∑l=1m‍μj∫-τ0‍kijl(s)∫t+st‍aijl(θ-s)×|xj*(θ)-xj(θ)|dθds.
Calculating the upper right derivation of Vi(t) along system (4) for i=1,2,…,n, we have
(52)D+Vi(t)=sign(xi*(t)-xi(t))×[-μiαi(t)(xi*(t)-xi(t))(xj*(t+θ)-xj(t+θ))ds+∑j=1n‍∑l=1m‍μjaijl(t)∫-τ0‍kijl(s)×(xj*(t+θ)-xj(t+θ))ds]+∑j=1n‍∑l=1m‍μj∫-τ0‍aijl(t-s)kijl(s)ds|xj*(t)-xj(t)|-∑j=1n‍∑l=1m‍μjaijl(t)∫-τ0‍kijl(s)×|xj*(t+θ)-xj(t+θ)|ds≤-μiαi(t)|xi*(t)-xi(t)|+∑j=1n‍∑l=1m‍μj∫-τ0‍aijl(t-s)kijl(s)ds|xj*(t)-xj(t)|.
Further, we define a Lyapunov function as follows:
(53)V(t)=∑i=1n‍Vi(t).
Calculating the upper right derivation of V(t), from (52) we finally can obtain, for all t≥0,
(54)D+V(t)≤-∑i=1n‍δi|xi*(t)-xi(t)|.
Integrating from0 to t on both sides of (54) and  by (50) produces
(55)V(t)+β∫0t‍(∑i=1n‍|xi*(s)-xi(s)|)ds≤V(0),t≥0,
then
(56)∫0t‍(∑i=1n‍|xi*(s)-xi(s)|)ds≤V(0)β,t≥0.
By the definition of V(t) and (53), we have
(57)∑i=1n‍μi|lnxi*(t)-lnxi(t)|≤V(t)≤V(0),t≥0.
Therefore, for i=1,2,…,n we have
(58)μi|lnxi*(t)-lnxi(t)|≤V(0),t≥0,
which, together with (49), leads to
(59)miexp{-V(0)μi}≤xi(t)≤Miexp{V(0)μi},i=1,2,…,n,
and, hence, ∑i=1n‍|xi*(t)-xi(t)|∈L1[0,+∞). From the boundedness of xi*(t) and (58), it follows that xi(t)(i=1,2,…,n) are bounded for t≥0. It is obvious that both xi(t) and xi*(t) satisfy the equations of system (4), then by system (4) and the boundedness of xi(t) and xi*(t), we know that the derivatives x˙i(t) and x˙i*(t) are bounded. Furthermore, we can obtain that x˙i*(t)-x˙i(t)(i=1,2,…,n) and their derivatives remain bounded on [0,+∞). Therefore ∑i=1n‍|xi*(t)-xi(t)| is uniformly continuous on [0,+∞). Thus, from (56), we have
(60)limt→+∞∑i=1n‍|xi*(t)-xi(t)|=0.
Therefor,
(61)limt→+∞(xi*(t)-xi(t))=0,i=1,2,…,n.
This completes the proof of Theorem 2.From the proof of Theorem2, on the existence and global attractivity of positive periodic solutions of system (5), we have the following result.Corollary 3.
Suppose that assumption(H1) holds and there exists a constant ρi>0(i=1,2,…,n) such that
(62)mint∈[0,ω]{ρiαi(t)-∑j=1n‍∑l=1m‍ρj∫-τ0‍ajil(t-s)kjil(s)ds}=:λi>0,i=1,2,…,n,
and the algebraic equation
(63)r-i-(α-i+a-ii)vi+∑j≠in‍a-ijvj=0,i=1,2,…,n,
has a unique positive solution. Then, system (5) has a positive ω-periodic solution which is globally attractive.


## 4. Two Examples

Example 4.
First, we consider the following delayed system:(64)x˙1(t)=x1(t)[2+cos(t)-(6+cos(t))x1(t)2+cos(t)3∫-τ0‍+2+cos(t)3∫-τ0‍k111(s)x1(t+s)ds+2+cos(t)3∫-τ0‍k121(s)x2(t+s)ds+2+cos(t)3∫-τ0‍k131(s)x3(t+s)ds],x˙2(t)=x2(t)[2+cos(t)-(5+cos(t))x2(t)3+cos(t)4∫-τ0‍+2+cos(t)7∫-τ0‍k221(s)x2(t+s)ds+3+cos(t)4∫-τ0‍k211(s)x1(t+s)ds+3+cos(t)4∫-τ0‍k231(s)x3(t+s)ds],x˙3(t)=x3(t)[2+cos(t)-(6+cos(t))x3(t)3+cos(t)5∫-τ0‍+3+cos(t)4∫-τ0‍k331(s)x3(t+s)ds+4+cos(t)5∫-τ0‍k311(s)x1(t+s)ds+3+cos(t)5∫-τ0‍k321(s)x2(t+s)ds].
Corresponding to system (4), n=3,  m=1,  ω=2π, by direct calculation, we can get
(65)σ1≈2,σ2≈1.6,σ3≈2.2,
and the following equations have unique positive solutions:
(66)a-11v1-a-12v2-a-13v3=r-1,a-22v1-a-21v2-a-23v3=r-2,a-33v1-a-31v2-a-32v3=r-3,
where
(67)v1≈0.1944,v2≈0.5625,v3≈0.8819.
It is clear that all the conditions of Theorem 2 hold. Hence, system (64) has a positive periodic solution which is globally attractive.Example 5.
Next, we consider the following delayed system:(68)x˙1(t)=x1(t)[2+cos(t)-5+4cos(t)20x1(t)+3+2cos(t)2∫-τ0‍k111(s)x1(t+s)ds+3+2cos(t)4∫-τ0‍k121(s)x2(t+s)ds+2+cos(t)2∫-τ0‍k131(s)x3(t+s)ds],x˙2(t)=x2(t)[2+cos(t)-41+40cos(t)100x2(t)+(2+cos(t))∫-τ0‍k221(s)x2(t+s)ds+9+6cos(t)8∫-τ0‍k211(s)x1(t+s)ds+3+2cos(t)4∫-τ0‍k231(s)x3(t+s)ds],x˙3(t)=x3(t)[2+cos(t)-5+4cos(t)20x3(t)+(4+cos(t))∫-τ0‍k331(s)x3(t+s)ds+5+4cos(t)8∫-τ0‍k311(s)x1(t+s)ds+6+3cos(t)8∫-τ0‍k321(s)x2(t+s)ds].
Corresponding to system (4), n=3,  m=1,  ω=2π, by direct calculation we can get
(69)σ1≈-5.2,σ2≈-6.1,σ3≈-7.2,
and the following equations have a unique positive solutions:
(70)a-11v1-a-12v2-a-13v3=r-1,a-22v1-a-21v2-a-23v3=r-2,a-33v1-a-31v2-a-32v3=r-3,
where
(71)v1=-0.1622,v2=0.7007,v3=1.2717.
Clearly, the conditions of Theorem 2 do not hold.
From Figure1 we can see that system (68) has no globally attractive positive periodic solution.Numerical simulation for system (68). Here, we take the initial value x0=(x10,x20,x30)=(1,1.5,2).Remark 6.
From these two examples, we can see that if the conditions of Theorem2 hold, then the system has a globally attractive positive periodic solution. If the conditions of Theorem 2 do not hold, then the system has no globally attractive positive periodic solution.


## 5. Conclusions

Mawhin’s continuation theorem is a powerful tool for studying the existence of periodic solutions of periodic high-dimensional time-delayed problems. When dealing with time-delayed problem, it is very convenient and the result is relatively simple. The most critical thing in the using of the theorem is the calculation of topological degree, that is, the condition (c) of the theorem.In this paper, motivated by  [2, 7] of Liu et al. we propose two classes of periodic N-species Lotka-Volterra facultative mutualism systems with distributed delays. By applying the continuation theorem of the coincidence degree theory developed by Gaines and Mawhin and the Lyapunov function method, we easily obtain sufficient conditions for the existence and global attractivity of positive periodic solutions of the system. From Theorem 2 and Corollary 3, we can see that the distributed time delays have effect on the existence and global attractivity of positive periodic solutions, and conditions (11) and (62) are very crucial to find the criteria for globally attractive positive periodic solutions. Further, the conditions (11) and (62) indicate that the undelayed intraspecific competition dominates the delayed intraspecific reproduction, and the intraspecific competition is more significant than the interspecific cooperation.



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*Source: 580185-2013-03-18.xml*