Permanence and existence of positive periodic solution for a multi-species cooperation system with continuous time delays, feedback control, and periodic external source

Nasertayoob Payam, Vaezpour, Vaezpour S

Description

In this paper, sufficient conditions which guarantee the existence of positive periodic solutions for a multi-species cooperation system are obtained. The permanence of solutions is studied. The proof is based on Schauder’s fixed-point theorem. Also, we give a illustrative example in order to indicate the validity of the assumptions. MSC: 54H99.

Sujets

• Periodic function
• Feedback

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 7
• Langue: English

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NasertayoobandVaezpourAdvancesinDifferenceEquations2012,2012:156
http://www.advancesindifferenceequations.com/content/2012/1/156
RESEARCH OpenAccess
Permanenceandexistenceofpositive
periodicsolutionforamulti-species
cooperationsystemwithcontinuoustime
delays,feedbackcontrol,andperiodic
externalsource
*PayamNasertayoob andSMansourVaezpour
*Correspondence:
[e-mail] Abstract
DepartmentofMathematics,
Inthispaper,sufficientconditionswhichguaranteetheexistenceofpositiveperiodicAmirkabirUniversityofTechnology
(Polytechnic),HafezAvenue,P.O. solutionsforamulti-speciescooperationsystemareobtained.Thepermanenceof
Box15914,Tehran,Iranisstudied.TheproofisbasedonSchauder’sfixed-pointtheorem.Also,we
giveaillustrativeexampleinordertoindicatethevalidityoftheassumptions.
MSC: 54H99
Keywords: Schauder’sfixed-pointtheorem;periodicsolution;multi-speciessystem;
feedbackcontrol
1 Introduction
May suggested the following system equations [] as the mathematical modeling of the
pairofmutualist:

x
x˙ =ρ x – –c x , 
a +b y 

y
y˙ =ρ y – –c y , 
a +b x 
where x(t), y(t) are densities of the species X, Y at time t,ρ , a , b , c , i=, are positivei i i i
constants. Subsequently, the nonautonomous version was argued by Cui and Chen [].
Cui in [] proposed the following generalization for the N-species cooperation system
withcontinuoustimedelays:

dx x (t)i i
=ρ (t)x (t) – –c (t)x (t) ,i i i indt a (t)+ b(t) V(θ)x(t+θ)dθi j j jj=,j= i –τ
x (t)=ϕ (t), forτ ≤t<, (.)i i

where V(θ)dθ=,j=,,...,N.j–τ
On the other hand, in the more realistic situation, the cooperation systems or
ecosystems are continuously perturbed via unpredictable forces. These perturbations are
gen© 2012 Nasertayoob and Vaezpour; licensee Springer. This is an Open Access article distributed under the terms of the
Creative
CommonsAttributionLicense(http://creativecommons.org/licenses/by/2.0),whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.NasertayoobandVaezpourAdvancesinDifferenceEquations2012,2012:156
Page2of15
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erallyresultsofthechangeinthesystem’sparameters.Inthelanguageofthecontroltheory,theseperturbationfunctionsmayberegardedascontrolvariables,andconsequently,
one should ask the question that whether or not an ecosystem can withstand those
unpredictableperturbationswhichpersistforafiniteperiodictime.Duringthelastdecade,
manyscholarsdidworksonthefeedbackcontrolecosystems.Someresultscanbefound
in [–] and the references therein. Chen, Lio, and Huang [] studied the dynamical
behavior of the following non-autonomous N-species cooperation system with continuous
timedelayandfeedbackcontrol:

dx x (t)i i
=ρ (t)x (t) – –c (t)x (t)i i i indt a (t)+ b (t) V (θ)x(t+θ)dθi ij ij jj=,j=i –τij

–d (t)u (t)x (t)–e (t)x (t) W (θ)u (t+θ)dθ,(.)i i i i i i i
–τi
dui
=–α (t)u (t)+β (t)x (t)+γ (t) U (θ)x (t+θ)dθ.)i i i i i i i
dt –ηi
wherex andu arethedensityofithcooperationspeciesandcontrolvariable,respectively.i i
a , b , c , d ,ρ ,α ,β,andγ ,areallcontinuousreal-valuedfunctionswhichareboundedi ij i i i i i i
aboveandbelowbypositiveconstants.Also,
  
V (θ)dθ = W (θ)dθ = U (θ)dθ=, i,j=,,...,N.i i i
–τ –τ –ηij i i
Veryrecently,ChenandXie[]obtainedasetofsufficientconditionsforpermanenceof
thesystemabove.Thestudywasbasedonanewintegralinequalityandtheresultsshowed
thatthefeedbackcontrolvariableshavenoinfluenceonthepermanenceofthesystem.In
the present paper, we study the sufficient condition for existence and permanence of the
positive periodic solutions of generalized version of the N-species cooperation system
(.),(.)whileforeachi,timevariationoftheithspeciesisaffectedbyexternalperiodic
sourceS (t),i.e.,i

dx x (t)i i
=ρ (t)x (t) – –c (t)x (t)i i i indt a (t)+ b (t) V (θ)x(t+θ)dθi ij ij jj=,j=i –τij

–d (t)u (t)x (t)–e (t)x (t) W (θ)u (t+θ)dθ –S (t), (.)i i i i i i i i
–τi
dui
=–α (t)u (t)+β (t)x (t)+γ (t) U (θ)x (t+θ)dθ.(.)i i i i i i i
dt –ηi
Our key tool is the following fixed-point theorem for a proper compact integral
operator on the convex subset of infinite dimension Banach space which is originally due to
Schauder[].
Theorem.(Schauder) LetX beaBanachspaceand
beaclosed,bounded,andconvex
subsetofX.If:
→
isacompactoperator,then hasatleastonefixedpointon
.
Besides,wealsoinvokethefollowingweakversionofArzela-Ascolitheorem[].NasertayoobandVaezpourAdvancesinDifferenceEquations2012,2012:156 Page3of15
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Theorem . (Arzela-Ascoli) Let {ξ (t)} be a sequence of real functions on [,T] whichn
isuniformlyboundedandequicontinuous.Then {ξ (t)}hasauniformlyconvergentsubse-n
quence.
Also,weset,
C = {ξ|ξ isacontinuousT-periodicfunctiononR},T
andforξ ∈C wedefine ξ=sup |ξ(t)|,T t∈[,T]

NB=C = =(ξ ,ξ ,...,ξ ):ξ ∈C ,i=,,...,N ,  N i TT
N N
andfor ∈B wedefine |(t)|= |ξ (t)|and = ξ .i B ii= i=
Clearly,(C , · )and(B, · )areBanachspaces.T B


T T
λ =exp ρ (t)dt , κ =exp α (t)dt ,i i i i
 
T T
A = β (t)+γ (t) dt, B = d (t)+e (t) dt,i i i i i i
 
T 
J (Q)= ρ (s) +c (s) ds, Q ∈R,i i ina (s)+Q b (s) i ijj=,j=i

I (Q)= + c .i in
inf [a (t)+Q b (t)]t∈[,T] i ijj=,j=i
Throughoutthispaper,weassumethat
(H ) a ,b ,c ,d ,e ,ρ ,α ,β,andγ ,i,j=,,...,N areallpositivecontinuousreal-valued i ij i i i i i i i
functions.
  
(H ) V (θ)dθ = W (θ)dθ = U (θ)dθ=,i,j=,,...,N. i i i–τ –τ –ηij i i
Thefollowingsectionisarrangedbasedontwomainsteps:Instep,weobtainsufficient
conditions with guarantee the existence of periodic solutions of each equation of system
(.)-(.)foreach ≤i ≤N.Intheproofofexistence,wewillusethemethodofGreen’s
functionsaccordingtoMokhtarzadehetal.[].Instep,wefollowChen[]toconstruct
a bounded, closed, and convex set in a product space and apply Schauder’s fixed-point
theorem.
2
Mainresults
Inthissection,weshallstudytheexistenceofperiodicsolutionsofthemulti-speciessystem(.)-(.).Todothis,wetransformthissystemofcoupleequationsintooneintegral
equation. For each  ≤i ≤N, we introduce the following integral operator on the Ba-i
nachspace(C , · ),T
:C →C ,i T T
T 
(ξ)(t)= G (t,s) β (s)ξ(s)+γ (s) U (θ)ξ(s+θ)dθ ds,(.)i i i i i
 –τiNasertayoobandVaezpourAdvancesinDifferenceEquations2012,2012:156 Page4of15
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Thekerneloftheintegraloperator(.)isinfacttheGreen’sfunctionofEq.(.)andis
givenby
⎧
T exp( α (θ)dθ) s⎪ i⎨ exp( α (θ)dθ),  ≤s ≤t ≤T,iT texp( α (θ)dθ)–iG (t,s)=i s⎪⎩ exp( α (θ)dθ),  ≤t ≤s ≤T,T itexp( α (θ)dθ)–i
T
where α =;see[].i
T
Lemma . Let  ≤ i ≤Nand α , β , γ,andU are belong to C as well as α =.i i i i T i
Supposethat u isacontinuousrealfunctionsuchthatforsomex ∈C , x =u.Thenui i T i i i
i
isaT-periodicsolutionofEq.(.).
TheproofofLemma.issimilartotheproofofLemma.of[].
AccordingtoLemma.,itmaybededucedthattheexistenceproblemofT-periodicsolutionofsystem(.),(.)isequivalenttothatoftheT-periodicsolutionof