. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 197, 427447 1996 ARTICLE NO. 0030 Perma

Ž.JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS197,427᎐4471996ARTICLE NO.0030Permanence in Kolmogorov-Type Systems of Nonautonomous Functional Differential EquationsBaorong Tang*and Yang Kuang†Department of Mathematics,Arizona State Uni¨ersity,Tempe,Arizona85287-1804Submitted by Mimmo IannelliReceived May14,1993;revised February23,1995In this paper we establish some general results on the permanence of the˙i n-species Kolmogorov-type nonautonomous functional differential system x sŽ.Ž.x t f t,x,i s1,2,...,n.These results are applied to the n-species Lotka᎐i i tŽ.Volterra-type systems with distributed delays,that is,when f t,x are lineari tfunctions of x.Our method consists of constructing n Liapunov-like functions,twhich can be regarded as boundary functions of some compact region inside thepositive cone in R n.ᮊ1996Academic Press,Inc.1.INTRODUCTIONA very basic and important ecological question associated with the study of multispecies population interactions is the long term coexistence of the involved populations.Such questions arise also in many other situations Žw x.see Hutson and Schmitt17.Mathematically,this is equivalent to the so-called persistence or permanence of the populations.Roughly speaking,Ž.Ž.we say a population x t is persistent if lim inf x t)0,and we say atªqϱsystem is persistent if all its populations are persistent.Uniform persistence requires that all involved populations will eventually have densities larger than some positive constant;if,in addition,these populations are bounded, then we say the system is permanent.Most of the existing persistence results are established for autonomous systems of ordinary differential equations.*Research partially supported by NSF Grant DMS9141550.†Research partially supported by NSF Grant DMS9102549.4270022-247X r96$18.00Copyrightᮊ1996by Academic Press,Inc.All rights of reproduction in any form reserved.TANG AND KUANG428The study of persistence and permanence of functional differential equations has received much attention recently.The general persistence theory for autonomous infinite dimensional systems is developed in Hale w x w x and Waltman 15.Freedman and Ruan 11established criteria for uni-form persistence in retarded functional differential equations.Burton and w x Hutson 2gave some persistence results for a class of Lotka ᎐Volterra-type autonomous functional differential equations with infinite delay.Persis-w x tence results are also given by Cao et al .5for two species autonomous w x Kolmogorov-type systems with single discrete delay,by Cao and Gard 6for n -dimensional Kolmogorov systems with single discrete delay,by Wang w x and Ma 22for two-dimensional Lotka ᎐Volterra-type prey ᎐predator sys-w x tems with discrete delays,by Wang 21for two competing species w x ŽLotka ᎐Volterra systems with discrete delays,by Kuang and Tang 19see w x .also Kuang 18,Chap.8for nonautonomous delay differential w x Kolmogorov-type population models and by Zanolin 23for delayed Kolmogorov competing species systems.For more details and more refer-ences on this subject,we refer to the recent survey papers of Hutson and w x w x Schmitt 17and Waltman 20.In this paper we present some results on the permanence of the n -dimensional Kolmogorov-type nonautonomous functional differential equationsx t s x t f t ,x ,i s 1,...,n , 1.1Ž.Ž.Ž.Ž.˙i i i t Žw x n .Ž.Ž.w x where x g C y ␶,0,R is defined by x ␪s x t q ␪,␪g y ␶,0,t t Žw x n .w x C s C y ␶,0,R is the space of continuous functions mapping y ␶,0n Ž.n to R with the uniform norm;and f s f ,...,f :R =C ªR is a given 1n function.We also present the applications of these results to the n -dimen-sional Lotka ᎐Volterra-type systems with distributed delayn m 0Žk .Žk .x s x t r t y a t x t q ␪d ␮␪,Ž.Ž.Ž.Ž.Ž.˙ÝÝH i i i i j k i j y ␶k s 1j s 1i s 1,...,n , 1.2Ž.and to the two-dimensional Lotka ᎐Volterra-type predator ᎐prey systems with distributed delay2m 0Ži .x s x t r t y a t x t q ␪d ␮␪,Ž.Ž.Ž.Ž.Ž.˙ÝÝH 111i j i 1j y ␶i s 1j s 1m 0Ž1.x s x t y r t q b t x t q ␪d ␮␪Ž.Ž.Ž.Ž.Ž.˙ÝH 2221j 12j y ␶j s 1m 0Ž2.yb t x t q ␪d ␮␪. 1.3Ž.Ž.Ž.Ž.ÝH 2j 22j y ␶j s 1FUNCTIONAL DIFFERENTIAL EQUATIONS 429Our method involves using n Liapunov-like functions to construct the Ž.boundary of the permanence region for system 1.1.Since it does not depend on the analysis of the behavior of the solutions on the boundary,it can be applied to nonautonomous systems.The key ideas are similar to w x those developed in 18,19,21,22.Ž.In the next section we establish permanence criteria for system 1.1.Ž.Ž.The applications of these criteria to systems 1.2and 1.3will be given in Section 3.2.MAIN RESULTSq Žw x n .Suppose ␶G 0is a given real number.We denote C s C y ␶,0,R q 55w x q 55with the uniform norm иon y ␶,0,that is,for ␾g C ,␾sϱϱÄ5Ž.5w x 455n max ␾s :s g y ␶,0,where иis a given norm in R .For anyw x n w .Ž.function x :y ␶,0ªR with ␶)0and any t g 0,␶,we define x иg q t q Ž.Ž.w x n ÄŽ.C as x ␪s x t q ␪for ␪g y ␶,0,where R s y s y ,...,y gt q 1n n 4Ž.Ž.R :y G 0,for i s 1,...,n .We denote x 0,␾␪the solution of system i t Ž.Ž.Ž.w x 1.1with initial condition x ␪s ␾␪,␪g y ␶,0.For convenience,we 0Ž.Ž.Ž.also write x t ,␾s x 0,␾t .Ž.We view x t ,␾in the above systems as population density at time t for i Ž.Ž.Ž.some species x .We therefore consider 1.1or 1.2or 1.3with initial i conditionsw x x ␪s ␾␪G 0,␪g y ␶,0,i s 1,...,n , 2.1Ž.Ž.Ž.i i Ž.Ž.where ␾i s 1,...,n is continuous.We denote ␾s ␾,...,␾.Hence i 1n ␾g C q .We assume that the global existence,uniqueness,and continuous Ž.dependence of solutions of system 1.1hold.Moreover,we assume that Ž.Ž.solutions of 1.1with initial conditions 2.1are nonnegative,and if Ž.Ž.␾0)0for some i ,then x t )0for all t G 0.Note that this assump-i i tion is natural from the biological point of view.Ž.D EFINITION 2.1.System 1.1is permanent if there exists a compactn Ž.region ⍀;Int R such that every solution of 1.1with initial conditionsq Ž.Ž.Ž.satisfying 2.1and ␾0)0i s 1,...,n will eventually enter and re-i main in region ⍀.Remark.This definition is equivalent to the usual definition of uniform persistence for systems of ordinary differential equations plus bounded-ness of solutions,and it is adopted as uniform persistence in several recent Žw x .papers on persistence of delay differential systems see,e.g.,18,19,22.Since our approach is rather geometric,we chose this definition.TANG AND KUANG430In real ecosystems,resources are limited and so are the populations that live on them.Therefore,we make the following assumption:Ž.A Dissipativity.There exists an M )0such that155x 0,␾F M for all t G T ␾q ␶Ž.Ž.t 1Ž.Ž.Ž.Ž.w x for some T ␾G 0,where x 0,␾␪s x t q ␪,␾,␪g y ␶,0.t Žn .ÄŽ.n In the following,we denote R s x s x ,...,x g R :x )0forq i 1n q j 4n ÄŽ.n 4j /i ,int R s x s x ,...,x g R :x )0,i s 1,...,n .q 1n q i D EFINITION 2.2.A differentiable function B :int R n ªint R n is calledq q a boundary function if it has the following property:For any M )0and any ␦)0there exists ␦)0,␦)0such that the 12following hold:Ž.Ä4n <<Ž.i Whenever j g 1,...,n ,x g int R ,x /M ,and B x F ␦,q j 1then x F ␦.j Ž.n <<Ž.ii Whenever x g int R ,x F M ,and B x G ␦for all j sq j 1,...,n ,then x G ␦for all j s 1,...,n .j 2Ž.ŽŽ.Ž..Ž.For example,let B x ,x s B x ,x ,B x ,x with B x ,x s 121122121123y 1Ž.y 1322x x and B x ,x s x x .It is easy to see that B :int R ªint R is 1221212q q differentiable.For any M )0and ␦)0,let ␦s ␦3M y 1,␦s ␦1r 2.12Ž.Ž.Then it is easy to check that properties i and ii in Definition 2.2are satisfied.Generally,we have the following.Ž.P ROPOSITION 2.1.The function B s B ,...,B with1n n ␤jkB x sx Ž.Łj j k s 1is a boundary function in the sense of Definition 2.2,if ␤)0,␤F 0for j j jk j /k ,j ,k s 1,...,n ,and one of the following two extra conditions is satisfied :Ž.a There exist numbers ␣,...,␣G 0such that1n n ␣␤)0᭙k s 1,...,n .Ýj jk j s 1Ž.b There exist numbers ␣,...,␣G 0such that1n n ␤␣)0᭙j s 1,...,n .Ýjk k k s 1FUNCTIONAL DIFFERENTIAL EQUATIONS 431n n Ž.Proof.Clearly B :int R ªint R is differentiable and property i q q Ž.Ž.easily follows without using the extra assumptions a or b .So we turn to Ž.Ž.property ii .First we assume the extra assumption a .Choose numbers ␣,...,␣G 0such that1n n ␣␤)0᭙k s 1,...,n .Ýj jk j s 1n <<Ž.Let x be in the interior of R ,x F M and B x G ␦for all j s 1,...,n .q j Thenn n n␣j ␣q иииq ␣␣␣Ý␣␤1n 1n j s 1j jk ␦s ␦иии␦F B x F x .Ž.ŁŁj k j s 1k s 1Ä4Let m g 1,...,n .Since x F M ,we havek ␦␣1q иииq ␣n F x Ýj s 1n ␣j ␤jm M Ýj s 1n␣␤.Łm j jk k /m This implies the assertion.Ž.Now we assume the extra assumption b .Choose numbers ␣,...,␣G 1n 0such thatn ␤␣)0᭙s 1,...,n .Ýjk k j k s 1n <<Ž.Let x be in the interior of R ,x F M and B x G ␦for all j s 1,...,n .q j Without loss of generality,we can assume that ␦F 1.Choose j such that0-x 1r ␣j F x 1r ␣k ᭙k s 1,...,n .j k Then,for this j and all m s 1,...,n ,␦F B x s x ␤j jx ␤jk F x ␤j j x ␤jk Ž␣k r ␣j .Ž.ŁŁj j k j j k /j k /j F x 1r ␣j Ýk s 1n ␤jk ␣k F x 1r ␣m Ýk s 1n ␤jk ␣k .j mHence,n y 1r ␣m x G ␦␤␣.Ým jk k k s 1TANG AND KUANG432Ä4We set ␣*s max ␣,...,␣and1n n ␥s min␤␣:j s 1,...,n .Ýjk k ½5k s 1Then ␥r ␣*1x G '␦.m 2ž/␦Ž.ŽŽ.Ž..q D EFINITION 2.3.We call V t ,␾s V t ,␾,...,V t ,␾,␾g C a 1n Ž.¨ector Liapuno ¨boundary function for system 1.1if there is a boundary Ž.ŽŽ.Ž..function B x s B x ,...,B x and n continuous functions G :R =1n i q Ä4C ªint R such that for all t g R ,M )0and i s 1,2,...,n ,q 0-inf G t ,␾F sup G t ,␾-q ϱ,Ž.Ž.i i 55␾F M 55␾F Mand all t ,s g R ,there is an ␣)0,such thatinf G t ,␾Ž.5␾5F M i )␣.sup G s ,␾Ž.5␾5F M i Ž.Ž.In addition,for x s x 0,␾,the solution of 1.1with x s ␾,andt t 0V t ,x s B x t ,␾G t ,x ,Ž.Ž.Ž.Ž.i t i i t we have d˙<V t ,x J Vt ,x G V t ,x P x t ,␾,Ž.Ž.Ž.Ž.Ž.Ž1.1.i t i t i t i i dt Ž.0Ž.where P :0,ϱªR and there is some x )0such that P x )0i i i i whenever 0-x -x 0.i i We are now in a position to state and prove our main result.Ž.T HEOREM 2.2.Suppose that A holds .If there is a ¨ector Liapuno ¨Ž.ŽŽ.Ž..Ž.Ž.function V t ,x s V t ,x ,...,V t ,x for system 1.1,then system 1.1t 1t n t is permanent .Proof.In the following we want to construct a permanent region ⍀as required in Definition 2.1.We divide the proof into five steps.Step 1.The construction of a bounded subset ⍀;int R n .Let M q 1Ž.Ž.be defined as in A ,and let the vector Liapunov function V t ,x betFUNCTIONAL DIFFERENTIAL EQUATIONS 433Ž.defined as in Definition 2.3.First we choose a ␦g 0,1such that if 00-x -␦,theni 0P x G ⑀for some ⑀)0,i s 1,...,n . 2.2Ž.Ž.i i 00Ž.Ä4Consider ␦g 0,␦such that,for any given i g 1,...,n ,we have the10n 55Ž.following:whenever x is in the interior of R ,x F M ,and B x F ␦,q 1i 1Ž.then x )␦.Such a ␦exists because of property i of a boundary i 01function.Further,by the definition of a vector Liapunov boundary func-tion,we can choosemin G t ,␾Ž.5␾5F M i 10-␦-␦for all t ,s g R ,i s 1,...,n .21max G x ,␾Ž.5␾5F M i 1We can now define a subset of R n :q ⍀s x s x ,...,x g R n :B x G ␦,x F M ,i s 1,...,n .Ä4Ž.Ž.1n q i 2i 1Ž.n Clearly the definition of B x yields that ⍀;int R .q In the following,we prove that ⍀is the desired permanent region.Ä4Step 2.We claim that the following holds for any i g 1,...,n ,5Ž.5ŽŽ..ŽŽ..t G 0.If x t ,␾F M for t G t and B x t ,␾G ␦,then B x t ,␾00i 01i G ␦for t G t .20Suppose this is not true;then there is some t )t such that 10ŽŽ..ŽŽ..ÄB x t ,␾)␦.Since ␦-␦and B x t ,␾G ␦,the set t :t )t ,i 1221i 010ŽŽ..4Ž..ˆB x t ,␾s ␦is not empty.Notice also that if B x ,t ,␾s ␦with i 2i 2ÄŽŽ..4ˆˆt )t ,then the set t :t )t )t ,B x t ,␾s ␦is not empty.Let00i 1t s min t :t )t ,B x t ,␾s ␦,Ä4Ž.Ž.30i 2t s max t :t -t -t ,B x t ,␾s ␦.Ä4Ž.Ž.203i 1ThenB x t ,␾s ␦,B x t ,␾s ␦Ž.Ž.Ž.Ž.i 32i 21Ž.and for t g t ,t ,23␦-B x t ,␾-␦.2.3Ž.Ž.Ž.2i 1Ž.Ž.By the choice of ␦and property i in Definition 2.2, 2.3yields that1x t ,␾-␦for t g t ,t . 2.4Ž.Ž.Ž.i 023The choice of ␦and the definition of a vector Liapunov boundary 0˙Ž.function imply that Vt ,x )0and so i t V t ,x )V t ,x .Ž.Ž.i 3t i 2t 32TANG AND KUANG434But V t ,x s B x t ,␾G t ,x Ž.Ž.Ž.Ž.i 3t i 3i 3t 33F B x t ,␾max G t ,␾Ž.Ž.Ž.i 3i 355␾F M 1s ␦max G t ,␾Ž.2i 355␾F M 1F ␦minG t ,␾Ž.1i 255␾F M 1s B x t ,␾min G t ,␾Ž.Ž.Ž.i 2i 255␾F M 1F B x t ,␾G t ,x Ž.Ž.Ž.i 2i 2t 2s V t ,x ,Ž.i 2t 2a contradiction which finishes Step 2.n ÄŽ.Step 3.By Step 2and noticing that ⍀s l x s x ,...,x g i s 11n n Ž.4R :B x G ␦,x F M ,j s 1,...,n ,we obtain that if there is some q i 2j 1ŽŽ..Ž.t G 0such that B x t ,␾G ␦for all i s 1,...,n ,then x t ,␾g ⍀0i 01for t G t .0Ž.Step 4.We claim that there exists ␦s ␦␾)0and some t #G 033such thatx t ,␾)␦,for t G t #,i s 1,...,n .Ž.i 3Ž.Ž.Clearly,this claim,together with assumption A imply that V t ,x i t Ž.i s 1,...,n are bounded.The following is the proof of this claim.5Ž.5ŽŽ..Let t #G 0be such that x t ,␾F M for t G t #and B x t #,␾s i X Ä4␦)0,i s 1,2,...,n .Define ␦s min ␦,i s 1,2,...,n .Then 1i 11i ŽŽ..X B x t #,␾G ␦,i s 1,2,...,n .Following Step 2,we can find a i 1X X ŽX .ŽŽ..X ␦,␦g 0,␦,such that B x t ,␾G ␦for t G t #and i s 1,2,...,n .221i 2Ž.By property ii of Definition 2.2,we see that there is a desired ␦)03Ž.such that x t ,␾G ␦for t G t #,i s 1,2,...,n .Note that this ␦de-i 33pends on ␾.Ž.Step 5.We claim that the solution x t ,␾enters and stays in ⍀for all large time t .We need to consider two cases:Ž.ŽŽ..a there is a t G 0such that B x t ,␾G ␦for all i s 1,...,n ;0i 01then step 3implies the claim,otherwiseFUNCTIONAL DIFFERENTIAL EQUATIONS 435Ž.Ž.b x t ,␾will remain in S _⍀for all large time t ,where1S s x s x ,...,x g R n :x F M ,i s 1,...,n ,Ä4Ž.1n q i 1⍀s x s x ,...,x g R n :B x G ␦,x F M ,i s 1,...,n .Ä4Ž.Ž.11n q i 1i 1Ä4In this case we first claim that there is no i g 1,...,n such thatB x t ,␾-␦for all large t .Ž.Ž.i 1Ä4ŽŽ..Otherwise,there is an i g 1,...,n such that B x t ,␾-␦for all i 1large t .The choice of ␦implies that1x t ,␾-␦for all large tŽ.i 0and by the choice of ␦,we have0dV t ,x G ⑀V t ,x ,Ž.Ž.i t 0i t dtwhich implies thatV t ,x ªq ϱas t ªq ϱ,Ž.i t Ž.Ž.contradicting the fact that V t ,x is bounded Step 4.This proves the i t above claim.Hence,there is a t )0such that1B x t ,␾G ␦.Ž.Ž.111Using Step 2,we see thatB x t ,␾G ␦for t G t .Ž.Ž.121For this t )0,the above claim implies that there is a t G t ,such that121B x t ,␾G ␦.Ž.Ž.221Using Step 2,we see thatB x t ,␾G ␦for t G t .Ž.Ž.222Repeating the above procedure,after n times,we arrive at some t )00such thatB x t ,␾G ␦for t G t ,i s 1,...,n .Ž.Ž.i 20This implies thatx t ,␾g ⍀for t G t ;Ž.0TANG AND KUANG436ŽŽ..Ž.i.e.,B x t ,␾G ␦for all t G t and i s 1,...,n .Property ii in Defini-i 22ˆtion 2.2yields a ␦)0which is independent of initial data such that ˆŽ.x t ,␾G ␦for t G t .This proves the theorem.i 0Ž.For the two-dimensional system,i.e.,when n s 2in system 1.1,the Ž.following result improves Theorem 2.2.Specifically,if B x is a boundary Ž.Ž.function,then it satisfies the properties i ᎐iii in the following theorem,Ž.Žy 1.but not the other way around.For example,B x ,x s x x ,x x 121212Ž.Ž.satisfies the properties i ᎐iii below,but it is not a boundary function.Ž.T HEOREM 2.3.Let n s 2and assume that A is true .Suppose that thereis a differentiable function B :int R 2ªint R 2with the following property :q q ˜ˆFor any M )0and any ␦,␥,⑀)0,there exists ␦)0,␦)0,␥)0such˜that the following hold :2˜Ž.55Ž.i Whene ¨er x g int R ,x F M ,and B x F ␦,then x F ⑀.q 11Ž.255Ž.Ž.ii Whene ¨er x g int R ,x F M ,B x G ␥and B x F ␥,then˜q 12x F ⑀.2Ž.255Ž.Ž.iii Whene ¨er x g int R ,x F M ,and B x G ␦,B x G ␥,then ,q 12ˆˆx G ␦,x G ␦.12Ž.Ž.Ž.Ž.Assume further that V t ,x ,y s B x ,y G t ,x ,y i s 1,2satisfy the i t t i i t t ŽŽ.conditions of Definition 2.3excluding the requirement that B x is a bound -.Ž.ary function .Then system 1.1is permanent .Proof.In the following we construct a permanent region as required in Definition 2.1.We again divide the proof into five steps.Step 1.The construction of a bounded subset ⍀;int R 2.Let M q 1Ž.Ž.be defined as in A and assume that the vector function V t ,x ,y st t ŽŽ.Ž..V t ,x ,y ,V t ,x ,y satisfies the conditions of the theorem.1t t 2t t Ž.First we choose a ␦g 0,1such that if 0-x -␦,then0i 0P x G ⑀for some ⑀)0;i s 1,...,2.Ž.i 100Ž.Choose ␦g 0,␦such that,whenever x is in the interior of 10255Ž.R ,x F M ,and B x F ␦,then x -␦.Such a ␦exists because of q 111101Ž.property i in the theorem.Choose ␦such that2inf G t ,␾Ž.5␾5F M i 10-␦-␦for all t ,s g R .21sup G s ,␾Ž.5␾5F M i 1Ž.255We also choose ␥g 0,␦such that whenever x g int R ,x F M ,10q Ž.Ž.B x G ␦,and B x F ␥,then x -␦.The existence of ␥is assured 1221201Ž.by property ii of the theorem.We choose ␥such that2inf G t ,␾Ž.5␾5F M i 10-␥-␥for all t ,s g R .21sup G s ,␾Ž.5␾5F M i 1We now define a subset of R 2:q ⍀s x s x ,x g R 2:B x G ␦,B x G ␥,x F M ,i s 1,2.Ž.Ž.Ž.Ä412q 1222i In the following,we prove that ⍀is the desired permanent region.5Ž5Step 2.We claim that the following holds for t G 0.If x t ,␾0F M 0ŽŽ..ŽŽ..for t G t and B x t ,␾G ␦,then B x t ,␾G ␦for t G t .This 0101120claim follows easily from Step 2in the proof of Theorem 2.2.5Ž.5Step 3.We claim that the following holds for t G 0.If x t ,␾F M 0ŽŽ..ŽŽ..ŽŽ..for t G t ,B x t ,␾G ␦,B s t ,␾G ␥,then B x t ,␾G ␦and 010120112ŽŽ..B x t ,␾G ␥for t G t .220ŽŽ..By Step 2,we know that B x t ,␾G ␦for t G t .So,in the following,120we show thatB x t ,␾G ␥for t G t .Ž.Ž.220Suppose that this is not true.Then there is some t )t such that 10ŽŽ..ŽŽ..ÄB x t ,␾-␥.Since ␥-␥and B x t ,␾G ␥,the set t :t )2221201ŽŽ..4ŽŽ..ˆt ,B x t ,␾s ␥is not empty.Notice also that if B x t ,␾s ␥02222ÄŽŽ..4ˆˆwith t )t ,then the set t :t )t G t ,B x t ,␾s ␥is not empty.Let0021t s min t :t )t ,B x t ,␾s ␥,Ä4Ž.Ž.3022t s max t :t -t -t ,B x t ,␾s ␥.Ä4Ž.Ž.20321ThenB x t ,␾s ␥,B x t ,␾s ␥,Ž.Ž.Ž.Ž.232221Ž.and for t g t ,t ,23␥-B x t ,␾-␥. 2.5Ž.Ž.Ž.221Ž.Ž.By the choice of ␥and property ii in the theorem, 2.5yields that1x t ,␾-␦for t g t ,t .Ž.Ž.2023ŽŽ..By repeating the rest of Step 2after 2.4in the proof of Theorem 2.2,we shall arrive at a contradiction.This proves the claim.Indeed,we have just ŽŽ..ŽŽ..proved the following:if B x t ,␾G ␦for all t G t ,and B x t ,␾G 12020ŽŽ..␥,then B x t ,␾G ␥for t G t .1220Step 4.Repeating Step 4in the proof of Theorem 2.2.Ž.Step 5.We claim that the solution x t ,␾enters and stays in ⍀for all large time t .We need to consider two cases:Ž.ŽŽ..ŽŽ..Case a .There is a t G 0such that B x t ,␾G ␦,B x t ,␾G 010120␥.In this case,Step 3implies the claim.1Ž.Ž.ÄŽ.2Ž.Case b .x t ,␾will remain in ⌬_x s x ,x g R :B x G12q 1Ž.4ÄŽ.2␦,B x G ␥,x F M ,i s 1,2,where ⌬s x s x ,x g R :x :121i 112q i 4M ,i s 1,2.Now we need to consider three subcases:1Ž.ŽŽ..ŽSubcase b1.B x t ,␾-␦for all large t .The choice of ␦see 111.Step 1gives thatx t ,␾F ␦for all large t ,Ž.10ŽŽ..and the choice of ␦yields that P x t,␾G ⑀for all large t ,so we 0110haved V t ,x G P x t ,␾V t ,x Ž.Ž.Ž.Ž.1t 111t dtG ⑀V t ,x for all large t ,Ž.01t which implies thatV t ,x ªq ϱas t ªq ϱ,Ž.1t Ž.Ž.contradicting the fact that V t ,x is bounded Step 4.This shows that 1t Ž.Subcase b1is not true and so there is some t G 0such that0B x t ,␾G ␦.Ž.Ž.101ŽŽ..Hence,by Step 2,B x t ,␾G ␦for all large t .So we still have the 12following two subcases to consider.Ž.Subcase b2.There is a t )0such that,for t G t ,0055x t ,␾F M ,B x t ,␾G ␦,Ž.Ž.Ž.112andB x t ,␾G ␥for some t G t .Ž.Ž.21110ŽŽ..In this case,the proof of Step 3indicates that B x t ,␾G ␥for t G t ,221proving the claim.Ž.5Ž.5Subcase b3.There is a t )0such that,for t G t ,x t ,␾F M ,001B x t ,␾G ␦,B x t ,␾-␥.Ž.Ž.Ž.Ž.1221Ž.Ž.In this case,by the choice of ␦and ␥see Step 1,we have x t ,␾F ␦2120ŽŽ..for all large t .The choice of ␦yields that P x t ,␾G ⑀for all large 2220t .Hence,d V t ,x G P x t ,␾V t ,x Ž.Ž.Ž.Ž.2t 222t dtG ⑀V t ,x for all large t ,Ž.02t which implies thatV t ,x ªq ϱas t ªq ϱ,Ž.2t Ž.Ž.contradicting the fact that V t ,x is bounded Step 4.This finishes the 2t proof for Step 5.5Ž.5ŽŽ..Hence there is a t G 0such that x t ,␾F M ,B x t ,␾G 011ŽŽ..Ž.␦,B x t ,␾G ␥for t G t .Now property iii in the theorem yields2220ˆthat there is a ␦)0which is independent of initial data,such that ˆx t ,␾G ␦for t G t ;i s 1,2.Ž.i 03.APPLICATIONSIn this section we will give some applications of the main result.TheŽ.first application concerns system 1.2,i.e.,n mŽk .Žk .x t s x t r t y a t x t q ␪d ␮␪,Ž.Ž.Ž.Ž.Ž.Ž.˙ÝÝH i i i i j k i j y ␶k s 1j s 1i s 1,...,n .3.1Ž.We letmŽk .Žk .Žk .␣t sat y ␪d ␮␪,Ž.Ž.Ž.ÝH i i ji j y ␶j s 1Žk .Žk .␣s sup ␣t :t g 0,ϱ,Ž..Ä4i i Žk .Žk .␣s inf ␣t :t g 0,ϱ,Ž..Ä4i i r s inf r t :t g 0,ϱ,Ä4Ž..i i r s sup r t :t g 0,ϱ,i ,k s 1,...,n .Ä4Ž..i iDefine¡Ž1.Žn .¦␣иии␣11......Ž1.Žn .␣иии␣i y 1i y 1Ž1.Žn .␣иии␣A s3.2Ž.i i i Ž1.Žn .␣иии␣i q 1i q 1......Ž1.Žn .¢§␣иии␣nnT Ž.and R s r ,...,r ,r ,r ,...,r ,i s 1,...,n .˜i 1i y 1i i q 1n Ž.Note that if system 3.1is autonomous,then A s A ,R s R for i j i j i ,j s 1,...,n .We make the following assumptions.Ž.Žk .Ž.Ž.H1a ti ,k s 1,...,n ;j s 1,...,m are nonnegative bounded i j m Ži .Ž.Ž.Ž.continuous functions and Ýa t and r ti s 1,...,n are positive j s 1i j i bounded continuous functions which are also bounded away from zero.Žk .Ž.Žk .Žq .␶)0and ␮␪are nondecreasing functions satisfying ␮0y i j i j Žk .Žy .␮y ␶s 1;i ,k s 1,...,n ;j s 1,...,m .i j Ž.0Žy 1.H2A is nonsingular and x s A R )0,i s 1,...,n .Wei i i i i Žj .Žj .y 1ŽŽi ..assume ␤)0,␤F 0,j ,k s 1,...,n ,j /k ,where A s ␤.j j jk i jk n =n Ž.Ž.The assumption H1makes system 3.1a competitive system.Clearly,existence,uniqueness,and continuous dependence of solutions are assured w x by Theorems 2.2.1᎐2.2.3in Hale 4.Moreover,it is easy to show that solutions are nonnegative in their maximum intervals of existence,and if Ž.Ž.␾0)0for some i ,then x t )0in its maximum interval of existence.i i Žm Ži .Ž..It is easy to prove since Ýa t is bounded away from zero thej s 1i j following uniform boundedness result.L EMMA 3.1.There exists an M )0such that for ␾g C q ,there is a Ž.Ž.Ž.T ␾)0,such that x t ,␾F M for all t G T ␾,i s 1,...,n .i Ž.Ž.Proof.By H1,the first equation of 3.1yields that for t G 0,x t F r t x t Ž.Ž.Ž.˙111and sox t F r x t .Ž.Ž.˙111Hence for t q ␪G 0,␪F 0,r ␪1x t q ␪G e x t .Ž.Ž.11Ž.Again from the first equation of 3.1,it follows thatmŽ1.Ž1.x t F x t r t y a x t q ␪d ␮␪Ž.Ž.Ž.Ž.Ž.˙ÝH 1111j 11j y ␶j s 1mr ␪Ž1.1F x t r ya ed ␮␪x t .Ž.Ž.Ž.ÝH 11i j1j 1ž/y ␶j s 1m 0r ␪Ži .i ÄŽŽ..4Letting s max r r Ýa H e d ␮␪:i s 1,...,n we have thati j s 1i j y ␶i j x t F for all large time t .Ž.1Ž.Ž.Applying similar arguments to x t ,...,x t ,we get that for i s 2,...,n ,2n x t F Mfor all large time t ,Ž.i which completes the proof.Ž.Ž.Ž.T HEOREM 3.1.Suppose that system 3.1satisfies H1᎐H2.Then sys -Ž.tem 3.1is permanent .Ž.␤l 1Žl .␤ln Žl .Proof.Let B x s x иииx andl 1n nn mŽl .G t ,x s exp y ␤Ž.ÝÝÝl t lii s 1k s 1j s 1tŽk .Žk .=a s y ␪x s ds d ␮␪,Ž.Ž.Ž.H H i jk i j y ␶t q ␪V t ,x s B x t G t ,x ,l s 1,...,n .Ž.Ž.Ž.Ž.l t l l t Ž.It follows from H2thatŽl .Žl .␤r q␤r )0᭙l s 1,...,n ,Ýll l li i i /lwhich is equivalent tonŽl .␤r )0᭙l s 1,...,n .Ýli i i s 1Ž.Hence,assumption b in Proposition 2.1is satisfied with ␣s r .j j Ž.ŽŽ.Ž..By Proposition 2.1,B x s B x ,...,B x is a boundary function.1n Ž.Also G t ,x satisfies the conditions in Definition 2.3.l tWe have dV t ,x Ž.l t <Ž3.1.dtnn n mŽl .Žl .Žk .s V t ,x ␤r t y␤a t Ž.Ž.Ž.ÝÝÝÝl t li i li i j ½i s 1i s 1k s 1j s 1Žk .=xt q ␪d ␮␪Ž.Ž.H ki j y ␶nn mŽl .Žk .Žk .y ␤x t at y ␪d ␮␪Ž.Ž.Ž.ÝÝÝH li k i ji j y ␶i s 1k s 1j s 1Žk .Žk .y ␣t xt q ␪d ␮␪Ž.Ž.Ž.H i j ki j 5y ␶nn nŽl .Žl .s V t ,x ␤r t y␤Ž.Ž.ÝÝÝl t li i lii s 1i s 1k s 1mŽk .Žk .=a t y ␪d ␮␪x t Ž.Ž.Ž.ÝH Hi j i j k ž/y ␶y ␶j s 1nn nŽl .Žl .Žk .s V t ,x ␤r t y␤␣t x t Ž.Ž.Ž.Ž.ÝÝÝl t li ili i k i s 1i s 1k s 1nŽl .Žl .s V t ,x ␤r t q␤r t Ž.Ž.Ž.Ýl t ll l li i i s 1i /lnn nŽl .Žk .Žl .Žk .y ␤␣t x t y␤␣t x t Ž.Ž.Ž.Ž.ÝÝÝlll k li i k k s 1i s 1k s 1i /lnnŽl .Žl .Žl .Žk .G V t ,x ␤r q␤r y␤␣x t Ž.Ž.ÝÝl t ll l li i lll k i s 1k s 1i /lnnŽl .Žk .y ␤␣x t .Ž.ÝÝli i k i s 1k s 1i /ln Žl .Žl .0Ž.By H2,␤r q␤r s x ,Ýll l li i l i s 1i /lnŽl .Žk .Žl .Žk .␤␣q␤␣s 0Ýll lli i i s 1i /lif k /l and equals 1if k s l .Hence we have0˙Vt ,x G V t ,x x y x t ,␾.Ž.Ž.Ž.l t l t l l Ž.Ž0.Ž.Let P x s x y x .Then V t ,x is a Liapunov boundary function forl l l l l t Ž.3.1.By Lemma 3.1,Theorem 2.2yields the conclusion.Ž.Žk .Ž.Žk .Ž.Suppose that 3.1is autonomous,i.e.,a t s a ,r t s r ,i ,k si j i j i i 1,...,n ,j s 1,...,m ,then ␣Žk .s Ým ␣Žk .andi j s 1i j ␣Ž1.иии␣Žn .11....A s ..Ž1.Žn .␣иии␣nnŽ.T Ž.and r s r ,...,r .Then A s A i s 1,...,n ,where A is defined as 1n i i Ž.in 3.2.We assumeŽ.H2ЈA is nonsingular and Ax s r has a positive solution;i.e.,x 0s A y 1r )0.We also assume that ␤)0,␤F 0,i ,k s 1,...,n ,ii ik y 1Ž.i /k ,where A s ␤.ik n =n Ž.Ž.Ž.C OROLLARY 3.2.Suppose 3.1is autonomous .Assume H1andH2ЈŽ.are true .Then system 3.1is permanent .Ž.Ž.Proof.Clearly H2Јimplies H2as required in Theorem 3.1.E XAMPLE .Consider the system1914x s x 1y x t y ␶y x t y ␶y x t y ␶,Ž.Ž.Ž.˙11111212313415x s x 1y x t y ␶y x t y ␶y x t y ␶,,Ž.Ž.Ž.˙22121222323111x s x 1y x t y ␶y x t y ␶y x t y ␶,Ž.Ž.Ž.˙33131232333663where ␶G 0,i ,j s 1,2,3.We havei j 19781A s ,814104277143y 1y 1y 1As s ␤,Ž.y 15y 3ik 3=3y 1y 25Ž.T 0y 1Ž.T and r s 1,1,1.Hence x s A r s 1,1,2)0.Also ␤)0and ii ␤F 0,i ,k s 1,2,3,and i /k .Hence Corollary 3.2implies that the ik above system is permanent.Ž.Next we apply Theorem 2.3to system 1.3,i.e.,2mŽi .x t s x t r t y a t x t q ␪d ␮␪,Ž.Ž.Ž.Ž.Ž.Ž.˙ÝÝH 111i j i 1j y ␶i s 1j s 1mŽ1.x t s x t y r t q b t x t q ␪d ␮␪Ž.Ž.Ž.Ž.Ž.Ž.˙ÝH 2221j 12j y ␶j s 1mŽ2.yb t x t q ␪d ␮␪, 3.3Ž.Ž.Ž.Ž.ÝH 2j 22jy ␶j s 10Žk .Ž.where H d ␮␪s 1for i ,k s 1,2,y s 1,...,m .Lety ␶i j mmŽi .Ži .␣t sat y ␪d ␮,␣sbt y ␪d ␮␪,Ž.Ž.Ž.Ž.ÝÝH H 1i i j1j 2i i j2j y ␶y ␶j s 1j s 1␣s sup ␣t :t g 0,ϱ,r s sup r t :t g 0,ϱ,Ä4Ä4Ž..Ž..k i k i i i ␣s inf ␣t :t g 0,ϱ,r s inf r t :t g 0,ϱ,Ä4Ä4Ž..Ž..k i k i i i i ,k s 1,2.Ž.Ž.Ž.T HEOREM 3.3.Suppose a t ,b t i s 1,2;j s 1,...,m are nonneg -i j i j Ž.Ž.ati ¨e bounded continuous functions and r t i s 1,2are positi ¨e boundedi continuous functions which are also bounded away from zero .If␣)0,␣)0,r ␣y r ␣)0, 3.4Ž.1122121211Ž.then system 3.3is permanent .Proof.It is known that there is an M )0such that for ␾g C q ,there Ž.Ž.Ž.Ž.Žis a T ␾G 0,such that x t ,␾F M i s 1,2for t G T ␾see Lemmai。