Pullback permanence in a non-autonomous competitive Lotka-Volterra model

Pullback pe manence in a non-au onomous compe i i e Lo ka-Vol e a model J. A. Langa a, J. C. Robinson b,1, A. Su´a ez a,2 aDp o. Ecuaciones Di e enciales y An´alisis Num´e ico, Fac. Ma em´a icas, C/ Ta ia s/n, C.P. 41012, Uni . Se illa, Spain. bMa hema ics Ins i u e, Uni e si y o Wa wick, Co en y, CV4 7AL, U. K. Abs ac The goal o his wo k is o s udy in some de ail he asymp o ic beha iou o a non-au onomous Lo ka-Vol e a model, bo h in he con en ional sense (as → ∞) and in he “pullback” sense (s a ing a ixed ini ial condi ion u he and u he back in ime). The non-au onomous e ms in ou model a e chosen such ha one species will e en ually die ou , uling ou any con en ional ype o pe manence. In con as we in oduce he no ion o “pullback pe manence” and show ha his p ope y is enjoyed by ou model. This is no jus a ma hema ical a i ice, bu a he shows ha i we come ac oss an ecology ha has been e ol ing o a e y long ime we s ill expec ha bo h species a e ep esen ed (and hei numbe s a e bounded below), e en i he inal a e o one o hem is less happy. The main ools in he pape a e he heo y o a ac o s o non-au onomous di e en ial equa ions, he sub-supe solu ion me hod and he spec al heo y o linea ellip ic equa ions. Key wo ds: Non-au onomous di e en ial equa ions, compe i i e di usion sys em, pullback a ac o , pe manence. 1991 MSC: 35J55, 35B41, 35K57, 37L05, 92D25. Email add esses: langa@nume .us.es (J. A. Langa), [email p o ec ed]a wick.ac.uk (J. C. Robinson), sua ez@nume .us.es (A. Su´a ez). 1JCR is cu en ly a Royal Socie y Uni e si y Resea ch Fellow, and would like o hank he Socie y o hei suppo , along wi h Ibe d ola o hei help du ing his isi o Se ille. 2This wo k has been pa ially suppo ed by P ojec CICYT MAR98-0486 and BFM2000-0797. P ep in submi ed o Jou nal o Di e en ial Equa ions 30 Ma ch 2004 1 In oduc ion In his pape we analyze he long- ime beha iou o he non-au onomous com- pe i i e Lo ka-Vol e a sys em                    u −∆u=u(λ−a( )u−b ) in Ω ×(s, +∞), −∆ = (µ−c −du) in Ω ×(s, +∞), u= = 0 on ∂Ω×(s, +∞), u(s, x) = u0(x), (s, x) = 0(x) in Ω, (1) whe e Ω is a bounded domain o RN,N≥1, wi h a smoo h bounda y ∂Ω, b, c,da e posi i e cons an s, λ, µ ∈Rand 0 < a( )≤A. P oblem (1) models he in e ac ions be ween wo compe ing species inhabi ing a egion Ω: u(x, ) and (x, ) ep esen he popula ion densi ies a loca ion x∈Ω and ime . Mo eo e , we a e assuming ha Ω is comple ely su ounded by inhospi able a eas, because bo h popula ion densi ies a e subjec o homogeneous Di ichle bounda y condi ions. He e, he ope a o −∆ akes in o accoun he di usi i y o he species, λand µa e he g ow h a es o he species, band ddesc ibe he in e ac ion a es be ween he species and inally, a( ) and ca e he limi ing e ec s o c owding in each popula ion. The s a ing poin o his pape is he ollowing obse a ion, which o ms he basis o he ela i ely ecen heo y o non-au onomous a ac o s as de eloped by C auel e al. [10], Kloeden & Schmal uss [19], and Schmal uss [30]. Suppose ha x( ;s, x0) deno es he solu ion o some sys em a ime ha is equal o x0a ime s. Fo an au onomous sys em we always ha e x( ;s, x0) = x( −s; 0, x0) and so conside ing he ime asymp o ic beha iou as →+∞is exac ly he same as conside ing wha happens as s→ −∞. Howe e , in a non-au onomous sys em he ini ial ime is as impo an as he inal ime, and hese wo di e en ypes o “ ime asymp o ic beha iou ” a e no equi alen . We do no aim he e o asse he p imacy o one o hese app oaches o e he o he , bu a he o demons a e ha he “pullback” p ocedu e (conside ing he beha iou as s→ −∞) is a use ul ool ha can add o ou unde s anding o non-au onomous sys ems. Simila ideas a e applied o he o dina y di e en ial equa ion e sion o (1) in Langa e al. [21] o which mo e de ailed esul s a e possible. In popula ion dynamics, a basic ques ion is o de e mine whe he he wo species will su i e in he long e m. This has been o malized as he c i e ion o pe manence (see Hale and Wal man [12], Hu son and Schmi [16] and 2 e e ences he ein). The sys em (1) is said o be pe manen i o any posi i e ini ial da a u0and 0, he solu ion (u( , s;u0, 0), ( , s;u0, 0)) en e s in ini e ime in o a compac se s ic ly bounded away om ze o in each componen . In he au onomous case, ha is when a( ) = a > 0, esul s abou pe manence ha e been ob ained using a ious echniques. These esul s depend on he alue o λand µwi h espec o ce ain p incipal eigen alues o associa ed linea ellip ic p oblems. We need some no a ion in o de o s a e hese esul s. Gi en ∈L∞(Ω), we deno e by λ1( ) he p incipal eigen alue o he p oblem      −∆w+ (x)w=σw in Ω, w= 0 on ∂Ω. We w i e λ1:= λ1(0). On he o he hand, gi en γ, e ∈Rand e > 0, we deno e by w[γ,e] he unique posi i e solu ion o      −∆w=γw −ew2in Ω, w= 0 on ∂Ω. Obse e ha wis ela ed o he s a iona y solu ion when only one species is p esen . I is well-known ha w[γ,e]exis s i , and only i , λ1< γ, and w[ ,e]≡0 i λ1≥γ. On he o he hand, i λ≤λ1o µ≤λ1, hen one o he wo species (o bo h o hem) will be d i en o ex inc ion. This ex inc ion egion was enla ged by L´opez-G´omez and Sabina in [25] (Co olla y 4.5) o a egion in he (λ, µ)-plane delimi ed by he cu es λ=λ0(µ) and µ=µ0(λ). Howe e , i λand µsa is y λ > ϕ(µ) and µ > ψ(λ) (2) whe e ϕ(µ) = λ1(bw[µ,c]) and ψ(λ) = λ1(dw[λ,a]), hen (1) is pe manen (see Can ell e al. [2], [4], [5] and L´opez-G´omez [24]). We would like o poin ou ha λ=λ1(bw[µ,c]) and µ=λ1(dw[λ,a]) de ine wo cu es in he (λ, µ)- plane whose beha iou is analyzed in de ail in [2] and [24]. In Figu e 1 we ha e summa ized he au onomous case o pa icula alues o he pa ame e s. In his Figu e we ha e deno ed by P:= {(λ, µ) : λ,µsa is y (2)}and by E:= {(λ, µ) : λ < λ0(µ) o µ < µ0(λ)}. In he non-au onomous case, p e ious wo k ocuses on nonlinea i ies ha a e pe iodic in ime, o ha a e bounded by pe iodic unc ions. In he i s case, he spec al heo y s ill wo ks and simila esul s o he au onomous case can be ob ained, see Hess [14] and Hess and Laze [15]. The second case was s udied 3 λ1 E λ µ λ=ϕ(µ) λ=λ (µ) µ=ψ(λ) µ=µ (λ) 0 0 P λ1 Fig. 1. Au onomous case. E: ex inc ion egion, P: pe manen egion. by Can ell and Cosne [3]. In [3] he au ho s assume ha 0 < a0≤a( )≤A o all ≥0, and using a compa ison me hod, hey show ha i λand µsa is y λ > λ1+µb a0 and µ > λ1+λd c, hen (1) is pe manen (Co olla y 3.1 in [3]). In his wo k, we do no assume ha a( ) is bounded below by a posi i e cons an and in ac we a e mainly in e es ed in he case a( )→0 as →+∞. We p o e in his case ha he e is no bounded abso bing se o (1), and so he sys em is no “pe manen ” in any con en ional sense. In ac , we analyse he o wa d beha iou in ime o (1) in de ail and we show ha one o bo h species a e d i en o ex inc ion when λ < λ1o λ > ϕ(µ).(3) See Figu e 2 whe e we ha e ep esen ed his case. We ha e deno ed by E= {(λ, µ) : λ,µsa is y (3)}. The idea o pullback con e gence om he heo y o andom and non-au ono- mous a ac o s (c . C auel e al. [10], Kloeden and Schmal uss [19], Schmal uss [30]) allows us o ask (and answe ) o he ques ions abou he beha iou o ou model (1). In pa icula we de ine he e a no ion o pullback pe manence: we say ha (1) is pullback pe manen i he e exis s a ime-dependen amily o (bounded) abso bing se s ha a e bounded away om ze o in each compo- nen . This idea is no in ended o eplace he s anda d no ion o pe manence, bu a he o complemen i . This de ini ion has an in e es ing biological in- e p e a ion: i we a i e a an island on which wo species ha e al eady been 4 λ1λ µ λ=ϕ(µ) λ1 EE E Fig. 2. Fo wa d beha iou in ime when a( )→0 as → ∞. compe ing (acco ding o ou model) o a long ime hen we can gua an ee ha nei he species will ha e died ou (and hei numbe s a e bounded below in a uni o m way, no ma e how long his ecology has been unning). This is new in o ma ion, no a ailable by conside ing he beha iou as →+∞: indeed, one migh expec om he ine i abili y o ex inc ion as → ∞ ha such beha iou would no occu . We ge he e pullback ex inc ion i λ < λ1o µ < λ1. Mo eo e , assuming ha a( )→a0>0 as → −∞ and λand µsa is y λ > ϕ(µ) and µ > ψ(λ, a0),(4) whe e ψ(λ, a0) = λ1(dw[λ,a0]), hen (1) is pullback pe manen . We ha e sum- ma ized his in Figu e 3, whe e E={(λ, µ) : λ≤λ1o µ≤λ1}. In ac we can gi e a bi mo e in o ma ion abou he s uc u e o he pull- back a ac ing s a es (“ he non-au onomous a ac o ”) by using he o de - p ese ing p ope y o (1) (we de ine an app op ia e o de in sec ion 3, c . [15], o example): a esul due o Langa and Su´a ez [22] shows ha (1) possesses wo ajec o ies, maximal and minimal, ha a e globally s able om abo e and below espec i ely. Finally, we should men ion he use o skew p oduc lows (Hale [11], Sell [29]) in s udying non-au onomous p oblems, pa icula ly in he pe iodic, quasi- pe iodic, o almos pe iodic case. The idea is o cons uc an au onomous semi low S( ) on he p oduc space H× F, whe e His he na u al phase space whe e he dynamics ake place (he e he dynamics o uand ) and F is he hull (see [29]) o all he ime dependen e ms o he equa ion. P o- ided ha Fis compac in some app op ia e opology he gene al heo y o dissipa i e dynamical sys ems can be applied o s udy S( ) on he space 5 Pullback Pe manence λ µ λ λ1 1 λ=ϕ(µ) µ=ψ(λ, a ) 0 Fig. 3. Non-au onomous case. E: pullback ex inc ion egion. H× F. Howe e , wi h an en i ely gene al non-au onomous e m he e is no clea choice o opology on F ha will make i compac , a p ope y c ucial o his app oach 3. This is highligh ed in he heo y o a ac o s o non- au onomous equa ions de eloped by Chepyzho & Vishik (see, o example, [6] and [7]): while hei s onges esul s equi e almos pe iodici y, p ecisely in o de o ob ain a compac F, hey s udy gene al non-au onomous e ms wi hou appealing o skew p oduc lows using he concep s o a “ke nel” and “ke nel sec ions”, he la e co esponding exac ly o he ime slices A( ) o he non-au onomous a ac o whose de ini ion we ecall below. An ou line o his pape is as ollows: in Sec ion 2 we in oduce he con- cep o a p ocess, gi e he de ini ion o a non-au onomous a ac o and s a e condi ions ha gua an ee i s exis ence. In Sec ion 3 we s udy p ope ies o o de -p ese ing p ocesses and in pa icula ecall a esul abou hei s abil- i y. In Sec ion 4 we s udy in de ail a non-au onomous logis ic equa ion which go e ns he beha iou o one o he species in absence o he o he : his sec- ion plays a c ucial ole h oughou all ha ollows. In Sec ion 5 we analyse bo h he o wa ds and pullback beha iou o sys em (1), and inish in sec ion 6 wi h he exis ence o a non-au onomous a ac o o (1) and condi ions o 3Using uni o m con e gence on R equi es almos pe iodici y. An in e es ing ex- ension should be possible unde he assump ion ha he non-au onomous e ms enjoy a uni o m modulus o con inui y o e R, o hen he opology o uni o m con e gence on compac subse s o Rwill make Fcompac , c . Johnson & Kloeden [17], and he ecen monog aph by Chepyzho & Vishik [8]. 6 pullback pe manence. 2 Non-au onomous a ac o s In his sec ion we in oduce he de ini ions o a non-au onomous a ac o and o pullback pe manence. Le (X, d) be a comple e me ic space (wi h me ic d) and {S( , s)} ≥s, , s ∈R be a amily o mappings sa is ying: a) S( , s)S(s, τ)u=S( , τ)u, o all τ≤s≤ , u ∈X, b) S( , τ)uis con inuous in ,τand u. c) S( , ) is he iden i y in X o all ∈R. Such a map is called a p ocess. Usually S( , τ)uwill a ise as he alue o he solu ion o a non-au onomous equa ion a ime wi h “ini ial condi ion” u a ime τ. As ema ked in he in oduc ion, o an au onomous equa ion he solu ions only depend on −τ, and we can w i e S( , τ) = S( −τ, 0). Le Dbe a non-emp y se o pa ame e ized amilies o non-emp y bounded se s {D( )} ∈R. In pa icula , D( )≡B∈ D, whe e B⊂Xis a bounded se . In wha ollows, we will conside a ixed base o a ac ion Dand h oughou ou analysis he concep s o abso p ion and a ac ion will be e e ed o his ixed base. Fo A, B ⊂Xde ine he Hausdo semidis ances as, dis (A, B) = sup a∈A in b∈Bd(a, b) Dis (A, B) = in a∈Ain b∈Bd(a, b). De ini ion 1 a) Gi en 0∈R, we say ha K( 0)⊂Xis a ac ing a ime 0i o e e y {D( )} ∈ D lim τ→−∞ dis (S( 0, τ)D(τ), K( 0)) = 0. A amily {K( )} ∈Ris a ac ing i K( 0)is a ac ing a ime 0, o all 0∈R. b) Gi en 0∈R, we say ha B( 0)⊂Xis abso bing a ime 0i o e e y {D( )} ∈ D he e exis s T=T( , D)∈Rsuch ha S( 0, τ)D(τ)⊂B( 0), o all τ≤T. A amily {B( )} ∈Ris abso bing i B( 0)is abso bing a ime 0, o all 0∈R. No e ha e e y abso bing se a ime 0is a ac ing. As discussed in he in oduc ion, his no ion akes he inal ime as ixed 7 and mo es he ini ial ime backwa ds owa ds −∞. We a e no e ol ing one ajec o y backwa ds in ime, bu a he we conside he cu en s a e o he sys em (a he ixed ime 0) which would esul om he same ini ial condi ion s a ing a ea lie and ea lie imes. This is called pullback a ac ion in he li e a u e (c . [18], [19], [30]). De ini ion 2 Le {B( )} ∈Rbe a amily o subse s o X. This amily is said o be in a ian wi h espec o he p ocess Si S( , τ)B(τ) = B( ), o all (τ, )∈R2, τ ≤ . No e ha his p ope y is a gene aliza ion o he classical p ope y o an in a ian se o a semig oup. Howe e , in his case we ha e o de ine he in a iance wi h espec o a amily o se s depending on a pa ame e . De ini ion 3 The amily o compac se s {A( )} ∈Ris said o be he global non-au onomous (o pullback) a ac o associa ed o he p ocess Si i is in a ian , a ac s e e y {D( )} ∈ D ( o all 0∈R) and minimal in he sense ha i {C( )} ∈Ris ano he amily o closed a ac ing se s, hen A( )⊂C( ) o all ∈R. The gene al esul on he exis ence o non-au onomous a ac o s is a gene - aliza ion o he abs ac heo y o au onomous dynamical sys ems (Temam [32], Hale [11]): Theo em 4 (C auel e al. [10], Schmal uss [30]) Assume ha he e exis s a amily o compac abso bing se s. Then, he amily {A( )} ∈Rde ined by A( ) = ∪D∈DΛ(D, ) is he global non-au onomous a ac o , whe e Λ(D, )is he omega-limi se a ime o D≡ {D( )} ∈ D, Λ(D, 0) = ∩s≤ 0∪τ≤sS( 0, τ)D(τ). Using he pullback idea in oduced abo e we can now gi e he ollowing de - ini ion o “pullback pe manence”. As in [5] we suppose ha X=X0∪∂X0, whe e X0is open, and X0,∂X0a e in a ian wi h espec o he p ocess S. In ou applica ion, ∂X0will be he se o solu ions wi h a leas one componen iden ically ze o. De ini ion 5 We say ha a sys em has he p ope y o pullback pe manence (o ha i is pe manen in he pullback sense) i he e exis s a ime-dependen 8 amily o bounded se s U:R7−→ X, sa is ying a) U( )abso bs e e y bounded se D⊂X(c . De ini ion 1). b) Dis (U( ), ∂X0)>0 o all ∈R. Following De ini ion 3, we can de ine a global a ac o A+⊂X0 ha a ac s e e y bounded se in X0: i s exis ence ollows using Theo em 4. 3 O de -p ese ing non-au onomous di e en ial equa ions In his sec ion we de ine wha i means o a p ocess o be o de -p ese ing. Fo such a p ocess we can de e mine some o he s uc u e o he non-au onomous a ac o and p o e he exis ence o a minimal and maximal ajec o y on he a ac o wi h some pa icula s abili y p ope ies. De ini ion 6 We say ha he p ocess {S( , s) : X→X} ≥sis o de -p ese ing i he e exis s an o de ela ion ‘¹’ in Xsuch ha , i w1¹w2, hen S( , s)w1¹ S( , s)w2, o all ≥s. The nex de ini ion gene elizes he concep o equilib ia in Hess [14], (see also A nold and Chuesho [1] in he s ochas ic case and Chuesho [9] in he non- au onomous case unde s onge condi ions). De ini ion 7 Le Sbe an o de -p ese ing p ocess. We call he con inuous map w:R→Xa comple e ajec o y i , o all s∈R,we ha e S( , s)w(s) = w( ), o ≥s. F om (w,w) such ha w( )¹w( ), o all ∈R,we can de ine he “in e al” Iw w( ) = {w∈X:w( )¹w¹w( )}. The ollowing esul was p o ed in [22] and i gi es su icien condi ions o he exis ence o uppe and lowe asymp o ically s able comple e ajec o ies, and p o ides some in o ma ion abou he s uc u e o he non-au onomous a ac o . Theo em 8 Le Sbe an o de -p ese ing p ocess and A( )i s associa ed pull- back a ac o a ac ing ime-dependen amilies o se s in a base o a ac ion D. Le w, w ∈ D be such ha w( )¹w( ), o all ∈R,and assume ha A( )⊂Iw w( ),∀ ∈R. 9 PROOF. I λ<λ1, hen obse e ha λ1(−λ) = λ1−λ > 0. Hence, om (14) and P oposi ion 10 c) we ge ha u( , s;u0, 0)→0 as → ∞. Simila ly, when µ≤λ1we ge ha ( , s;u0, 0)→0 as → ∞. Now, we assume µ > λ1. Le δ > 0 be such ha µ > λ1+dδ. Fo such δ he e exis s 0∈Rsuch ha ku( , s;u0, 0)k∞< δ o any ≥ 0. On he o he hand, using he de ini ion o θ[q,b]we ob ain u=θ[λ−b ,a]and =θ[µ−du,c].(15) Then, by (14) and P oposi ion 10 a), we ha e θ[µ−dδ,c]≤θ[µ−du,c]= ≤θ[µ,c], o ≥ 0. I is su icien o apply P oposi ion 10 b) and he con inui y o he map 7→ w[ ,e].¤ The ollowing esul shows ha he sys em is no pe mamen when λand µ sa is y an easily e i iable condi ion. The sys em is no pe manen because one species (u) inc eases inde ini ely and d i es he o he o ex inc ion. We no e he e ha al hough unde he condi ion a( )→0 he equa ion is “asymp o ically au onomous” in he sense o Ma kus [26] (see also mo e e- cen wo ks by Thieme [33], Mischaikow e al. [27]) he gene al esul s ha a e a ailable o such sys ems a e no su icien ly de ailed o gi e us all he in o ma ion we need: o example, i is known ha i all he solu ions o he limi equa ion a e unbounded hen so a e he solu ions o he non-au onomous equa ion [26], bu we wish o show ha while one species g ows wi hou bound he o he is d i en o ex inc ion. P oposi ion 13 Suppose a( )→0as → ∞. I λ > λ1(bw[µ,c]), hen (u( , s;u0, 0), ( , s;u0, 0)) →(∞,0) as → ∞. Obse e ha w[µ,c]= 0 i µ≤λ1, so λ > λ1(bw[µ,c]) means λ > λ1when µ≤λ1. PROOF. Assume µ≤λ1, hen by P oposi ion 10 c) we ha e ha ≤θ[µ,c]→ 0 as → ∞. Mo eo e , since λ > λ1, we can ob ain ha λ−bθ[µ,c]> λ1 o ≥ 1. 16 Hence, λ1(−λ+bθ[µ,c])< λ1(−λ1) = 0, and so, by P oposi ion 10 d) θ[λ−bθ[µ,c],a]→ ∞, and he esul ollows by (14). Now, suppose µ > λ1and λ > λ1(bw[µ,c]). We de ine ε:= λ−λ1(bw[µ,c]) 2b Since ≤θ[µ,c]→w[µ,c]as → ∞, hen he e exis s εsuch ha o ≥ ε ≤w[µ,c]+ε, and so, by (15) u=θ[λ−b ,a]≥θ[λ−b(w[µ,c]+ε),a], o ≥ ε. Since, a( )→0 as → ∞, gi en δ∈(0,1] he e exis s δsuch ha o ≥ δ we ha e a( )≤δ, and so, u≥θ[λ−b(w[µ,c]+ε),a]≥θ[λ−b(w[µ,c]+ε),δ], ≥max{ ε, δ}.(16) Now, obse e ha λ1(−λ+bw[µ,c]+bε) = λ1(bw[µ,c])−λ+bε =−λ−λ1(bw[µ,c]) 2<0.(17) Taking accoun (16) and (17), a simila a gumen o he used in he p oo o P oposi ion 10 d) shows ha gi en a small posi i e σ > 0 he e exis s σsuch ha o ≥ σ, we ha e u≥Φ := λ−λ1(bw[µ,c]) 2δϕ1(−λ+b(w[µ,c]+ε)) −σ. (18) Taking σsuch ha 0< σ < λ−λ1(bw[µ,c]) 4≤λ−λ1(bw[µ,c]) 4δ(19) 17 we ge ha kuk∞≥ kΦk∞≥λ−λ1(bw[µ,c]) 4δ. Hence, i is su icien o ake δsu icien ly small in o de o show ha u app oaches in ini y. Finally, obse e ha by (18) we ge =θ[µ−du,c]≤θ[µ−dΦ,c], ≥ σ, and i we can ake µ < λ1(dΦ), by P oposi ion 10 b) we ob ain ha goes o 0. Bu , µ < λ1(dΦ) is equi alen o µ < λ1Ãλ−λ1(bw[µ,c]) 2δϕ1(−λ+b(w[µ,c]+ε))!−σ, which is ue by (19) and (7) aking δsu icien ly small. This comple es he p oo . ¤ 5.2 Pullback asymp o ic beha iou The nex wo esul s show “pullback” ex inc ion o some alues o λand µ. The i s one is simila o P oposi ion 12 and so we omi he p oo . P oposi ion 14 Suppose λ < λ1. a) I µ≤λ1, hen (u( , s;u0, 0), ( , s;u0, 0)) →(0,0) as s→ −∞. b) I µ > λ1, hen (u( , s;u0, 0), ( , s;u0, 0)) →(0, w[µ,c])as s→ −∞. He ea e , we deno e A:D(A)7→ C0(Ω) he linea ope a o associa ed o he Laplacian. P oposi ion 15 Gi en ∈R,λ > λ1and µ≤λ1, hen (u( , s;u0, 0), ( , s;u0, 0)) →(θ[λ,a]( , s;u0),0) as s→ −∞. PROOF. Since µ≤λ1, hen ≤θ[µ,c]→0 as s→ −∞. Now, gi en δ > 0 he e exis s sδsuch ha ( , s;u0, 0)≤δ o s≤sδ. 18 Hence, by (15), we ge θ[λ−bδ,a]≤θ[λ−b ,a]=u≤θ[λ,a], o s≤sδ, and so, θ[λ−bδ,a]−θ[λ,a]≤u−θ[λ,a]≤0, o s≤sδ. Thus, i su ices o p o e ha wδ:= θ[λ−bδ,a]−θ[λ,a]→0,as δ→0. (20) I is no ha d o p o e ha wδsa is ies (wδ) −∆wδ=λwδ−bδθ[λ−bδ,a]−a( )wδ(θ[λ−bδ,a]+θ[λ,a]). Now, i we deno e by gδ( , s) = λ−a( )(θ[λ−bδ,a]( , s;u0) + θ[λ,a]( , s;u0)) and w i ing wδ om he a ia ion o cons an s o mula, we ob ain wδ( , s;u0) = Z s e−A( − )(gδ( , s)wδ( , s;u0)−bδθ[λ−bδ,a]( , s;u0)) d , and so, since ° ° °e−A( − )° ° °op ≤1, we ge kwδ( , s;u0)k∞≤ Z s kgδ( , s)k∞kwδ( , s;u0)k∞d +bδ Z s kθ[λ−bδ,a]( , s;u0)k∞d , and by G onwall’s lemma we ob ain kwδ( , s;u0)k∞≤bδ Z s kθ[λ−bδ,a]( , s;u0)k∞d ·eR skgδ( ,s)k∞d .(21) On he o he hand, by P oposi ion 10 we ha e kθ[λ−bδ,a]( , s;u0)k∞≤ kθ[λ,a]( , s;u0)k∞≤ ( ) o s≤T( ), o some T( ) and ( ) independen o δ. Now, (20) ollows by aking δ o ze o in (21). ¤ The nex esul shows ha o a ixed inal ime 0, he posi i e solu ion o (1) is bounded away by posi i e unc ions o ssu icien ly small. 19 P oposi ion 16 Fix 0∈R. Assume ha in s∈(−∞, 0]a(s) = α( 0)>0, λ > λ1(bw[µ,c]),and µ > λ1(dw[λ,α( 0)]). Then, he e exis s0≤ 0and ei∈C0(Ω) posi i e unc ions (depending on 0), such ha o all s≤s0: u( 0, s;u0, 0)≥e1and ( 0, s;u0, 0)≥e2. PROOF. Since α( 0)≤a( )≤A o all ≤ 0, we ha e θ[λ,A]( 0, s;u0)≤θ[λ,a]( 0, s;u0)≤θ[λ,α( 0)]( 0, s;u0) o s≤ 0. Since λ > λ1(bw[µ,c]), µ > λ1(dw[λ,α( 0)]), we can choose ε > 0 su icien ly small such ha λ > λ1(b(w[µ,c]+ε)),and µ > λ1(d(w[λ,α( 0)] +ε)).(22) Fo such ε > 0, and by P oposi ion 10 b), we ob ain w[λ,A]−ε≤θ[λ,a]( 0, s;u0)≤w[λ,α( 0)] +ε o s≤s0, o some s0. Using again P oposi ion 10 a) and (14), we ge θ[µ−d(w[λ,α( 0)]+ε),c]≤ , o s≤s0. (23) On he o he hand, by P oposi ion 10 a) θ[λ−bθ[µ,c],A]≤θ[λ−bθ[µ,c],a]≤u and by pa b), w[µ,c]−ε≤θ[µ,c]( 0, s;u0)≤w[µ,c]+ε o s≤s0, and so, θ[λ−b(w[µ,c]+ε),A]≤u. (24) 20 Now, by P oposi ion 10 b), we ha e ha as s→ −∞, θ[µ−d(w[λ,α( 0)]+ε),c]→w[µ−d(w[λ,α( 0)]+ε),c], θ[λ−b(w[µ,c]+ε),A]→w[λ−b(w[µ,c]+ε),A]. P oposi ion 10 b), (22), (23) and (24) comple e he p oo . ¤ Assuming ha a( ) ends o a posi i e cons an as → −∞, we ob ain a simila esul o P oposi ion 16 bu whe e he condi ions on λand µdo no depend on . Co olla y 17 Assume a( )→a0>0as → −∞, o each ∈R in s∈(−∞, ]a(s) = α( )>0, λ > λ1(bw[µ,c])and µ > λ1(dw[λ,a0]). Then, o all ∈R, he e exis s0( )≤ and i∈C0(Ω) posi i e unc ions (depending on ), such ha o all s≤s0i holds: u( , s;u0, 0)≥ 1and ( , s;u0, 0)≥ 2. PROOF. Since µ > λ1(dw[λ,a0]) and om he con inui y o he map e7→ w[λ,e], he e exis s ε > 0 such ha µ>λ1(dw[λ,a0−ε]). On he o he hand, since a( )→a0as → −∞, he e exis s T∈Rsuch ha o all ≤T, a0−ε≤α( )≤a( )≤A. Then, o any 0≤Twe ha e ha µ > λ1(dw[λ,α( 0)]), and so by P oposi ion 16, we ge ha he e exis wo posi i e unc ions ei such ha u( 0, s;u0, 0)≥e1and ( 0, s;u0, 0)≥e2. Fu he mo e, o all ≥ 0we ha e u( , s;u0, 0) = u( , 0;u( 0, s;u0, 0), ( 0, s;u0, 0)) om which, by he s ong maximum p inciple, we ob ain he esul . ¤ 21 6 Exis ence o a non-au onomous a ac o and pullback pe ma- nence o he Lo ka-Vol e a compe i ion model We de ine X:= C0(Ω) ×C0(Ω) and he ollowing p ocess in X: o , s ∈R, ≥s, S( , s) : X7→ X;S( , s)(u0, 0) = (u( , s;u0, 0), ( , s;u0, 0)), whe e (u( , s;u0, 0), ( , s;u0, 0)) is he unique posi i e solu ion o (1) o u0, 0∈P. Mo eo e , in Xwe de ine he ollowing o de : gi en (u1, 1),(u2, 2)∈ X, (u1, 1)¹(u2, 2) i , and only i , u1≤u2and 1≥ 2, whe e “≤” is he o de de ined by Pin C0(Ω). I is well-known, see [15], ha S( , s) is an o de -p ese ing p ocess, ha is, i (u1, 1)¹(u2, 2), hen S( , s)(u1, 1)¹S( , s)(u2, 2). Mo eo e , we conside he no m |(u, )|∞= kuk∞+k k∞in X. In he nex wo sec ions we will p o e he exis ence o a non-au onomous a ac o o (1). 6.1 Abso bing se in X Le D⊂Xbe bounded, i.e., supd∈D|d|∞≤M, o M > 0,and (u0, 0)∈D. By (14) and P oposi ion 10 e), he e exis s T( , u0, 0)∈Rsuch ha ku( , s;u0, 0)k∞≤° ° °θ[λ,a]( , s;u0)° ° °∞≤ λ( ) o s≤T( ), (25) whe e λ( ) = 2eλ R −∞ eλτ a(τ) dτ. Simila ly, k ( , s;u0, 0)k∞≤ µ( ) o s≤T( ), (26) whe e µ( ) = 2eµ cR −∞ eµτ dτ=2µ c. 22 Clea ly, his means ha he ball in Xwi h adius 1( ) = λ( ) + µ( ), BX(0, 1( )),is abso bing o he p ocess S( , s). 6.2 Abso bing se in C1 0(Ω) ×C1 0(Ω) In o de o ob ain a amily o abso bing se s in C1 0(Ω) we need he ollowing esul om [28], see also Lemma 3.1 in [4]. He e, o a Banach space Y, Y β will deno e he usual ac ional powe spaces wi h no m |·|β. Recall ha A: D(A)7→ C0(Ω) is he linea ope a o associa ed o he Laplacian. Lemma 18 The ope a o Agene a es an analy ic semig oup on Y=Ck 0(Ω) o k= 0,1.Mo eo e Yβ,→Ck+q 0(Ω) o q= 0,1and 2β > q. Gi en D⊂Xbounded, we de ine o ≥s h( , s) = λu( , s;u0, 0)−a( )u2( , s;u0, 0)−bu( , s;u0, 0) ( , s;u0, 0). Then, w i ing u om he a ia ion o cons an s o mula, we ob ain u( , s;u0, 0) = e−A( −s)u0+ Z s e−A( − )h( , s) d . Hence, be ween −1 and , we ge u( , s;u0, 0) = e−Au( −1, s;u0, 0) + Z −1 e−A( − )h( , s) d . Hence, |u( , s;u0, 0)|β=° ° °Aβu( , s;u0, 0)° ° °∞≤° ° °Aβe−A° ° °op ku( −1, s;u0, 0)k∞+ sup ∈[ −1, ]kh( , s)k∞R −1° ° °Aβe−A( − )° ° °op d . Now, using he es ima e ° ° °Aβe−A( − )° ° °op ≤Cβ( − )−βe−δ( − ) 23 o some cons an s Cβ, δ > 0 (c . Hen y [13]), and he es ima es (25) and (26), we ob ain he exis ence o M( ) and T0( ) such ha |u( , s;u0, 0)|β≤M( ) o all s≤T0( ), wi h β < 1−ε, and any ε∈(0,1).Applying now Lemma 18 wi h q= 1 and β > 1/2,we ob ain ku( , s;u0, 0)kC1≤R1(D, ) o all s≤T0( ). Simila ly, i can be p o en ha k ( , s;u0, 0)kC1≤R2(D, ) o all s≤T0( ), o some R2(D, ), and so he ball in C1 0(Ω) ×C1 0(Ω), B(0, R( )) is abso bing in C1 0(Ω) ×C1 0(Ω), o R( ) = R1( ) + R2( ), whe e again we ha e used he no m |(u, )|C1(Ω) =kukC1(Ω) +k kC1(Ω) in C1 0(Ω) ×C1 0(Ω). We can epea he a gumen aking Y=C1 0(Ω) and Da bounded se in Y×Y. In his case, using Lemma 18 again, we ob ain ku( , s;u0, 0)kC2≤N(D, ) o all s≤T1( ), and hence, he exis ence o an abso bing se ha is bounded in C2 0(Ω)×C2 0(Ω), and so compac in X. Analogously we can show he exis ence o he global a ac o A+a ac ing e e y bounded se in X0. 6.3 On he s uc u e o he pullback a ac o and pullback pe manence In his sec ion we apply he esul s o Sec ion 3 o ou model. We ake w( ) = (0, µ( )) and w( ) = ( λ( ),0). Fi s ly, obse e ha w( )¹w( ). On he o he hand, by (25) and (26) i ollows ha A( )⊂Iw w( ). 24 Finally, we de ine he base o a ac ion in ou model as D:= {w:R7→ Xcon inuous, such ha , lim s→−∞ eγs kw(s)k∞ = 0} whe e γ= min{λ, µ}. No e, ha gi en w= (u, )∈ D, lim s→−∞ dis (S( , s)(u(s), (s)),A( )) = 0.(27) Indeed, we ha e ha o ssmall enough ku( , s;u(s), (s))k∞≤ kθ[λ,a]( , s;u(s))k∞≤eλ eλs ku(s)k∞ +R seλτ a(τ) dτ≤ λ( ). Mo eo e , i is clea ha (w, w)∈ D. So, applying Theo em 8, he e exis comple e ajec o ies w∗(minimal) and w∗(maximal) ha a e s able in he sense o Theo em 8. In a simila way, o A+we can also apply Theo em 8 o w( ) = ( 1( ), µ( )), w( ) = ( λ( ), 2( )), so ha , o s ic ly posi i e ini ial da a, he non-au onomous a ac o is bounded abo e and below by s ic ly posi i e bounds. Finally, we can con- clude he pullback pe manence o ou model. Theo em 19 Assume ha a( )→a0>0as → −∞, o each ∈R in s∈(−∞, ]a(s) = α( )>0, λ > λ1(bw[µ,c])and µ > λ1(dw[λ,a0]). Then (1) is pe manen in he pullback sense. PROOF. We w i e X=X0∪∂X0, whe e X0= (in P)2and ∂X0=X X0. The pe manence ollows wi h U( ) = {w∈X: ( 1( ), µ( )) ¹w¹( λ( ), 2( ))}, whe e 1, 2a e de ined in Co olla y 17 and λand µin (25) and (26), espec- i ely. By Sec ion 6.1, U( ) is abso bing and by Co olla y 17 Dis (U( ), ∂X0)> 0. This comple es he p oo . ¤ 25