Forwards and pullback behaviour of a non-autonomous Lotka-Volterra system

Fo wa ds and pullback beha iou o a non-au onomous Lo ka-Vol e a sys em Jos´e A. Langa†, James C. Robinson‡, An onio Su´a ez† †Depa amen o de Ecuaciones Di e enciales y An´alisis Num´e ico, Uni e sidad de Se illa, Apdo. de Co eos 1160, 41080-Se illa, Spain. ‡Ma hema ics Ins i u e, Uni e si y o Wa wick, Co en y, CV4 7AL, U.K. E-mail: [email p o ec ed]; [email p o ec ed]; [email p o ec ed] Abs ac . Lo ka-Vol e a sys ems ha e been ex ensi ely s udied by many au ho s, bo h in he au onomous and non-au onomous cases. In p e ious pape s he ime asymp o ic beha iou as → ∞ has been conside ed. In his pape we also conside he “pullback” asymp o ic beha iou which oughly co esponds o obse ing a sys em “now” ha has al eady been e ol ing o a long ime. Fo a compe i i e sys em ha is asymp o ically au onomous bo h as → −∞ and as →+∞we show ha hese wo no ions o asymp o ic beha iou can be e y di e en bu a e bo h impo an o a ull unde s anding o he dynamics. In pa icula he e a e pa ame e anges o which, al hough one species dies ou as → ∞, he e is a dis inguished ime-dependen coexis en s a e ha is a ac ing in he pullback sense. AMS classi ica ion scheme numbe s: P ima y 37G10, 37G35; Seconda y 34D05, 34D23. Submi ed o: Nonlinea i y A non-au onomous Lo ka-Vol e a sys em 2 1. In oduc ion In his pape we conside a non-au onomous compe i i e Lo ka-Vol e a sys em o wo species uand , (˙u=u(λ−a( )u−b )u(s) = u0>0 ˙ = (µ−c −du) (s) = 0>0,(1) whe e a( ) is con inuous, lim s→+∞a(s) = 0 < a( )< A = lim s→−∞ a(s) o all ∈R and b, c, d > 0. We will also only conside he case λ, µ > 0, since when ei he o hese pa ame e s a e nega i e he beha iou is ela i ely simple. No e ha ou model equa ion is asymp o ically au onomous†bo h as →+∞and as → −∞. We will e e o sys em (1) as E[a( )]: such a no a ion makes i easie o discuss compa isons o solu ions o (1) wi h he solu ions o a ious au onomous sys ems E[a] wi h a( ) eplaced by a cons an a( he beha iou o such sys ems is well unde s ood, e.g. Mu ay [20], and we ecall his as equi ed in wha ollows). Since he uand axes a e in a ian and solu ions o (1) a e unique he posi i e cone P={(u, ), u ≥0, ≥0}and i s in e io P={(u, ), u > 0, > 0}a e in a ian se s. I is he e o e consis en o conside only posi i e solu ions, which is also na u al in he ligh o he ecological in e p e a ion o (1) as a model o wo compe ing species. Va ious au ho s ha e conside ed non-au onomous e sions o he equa ion concen a ing on he asymp o ic beha iou as →+∞. Howe e , he e is ano he poin o iew which, al hough equi alen in he au onomous case, allows o o he a gumen s in he non-au onomous si ua ion. We s udy he solu ions in e ms o he co esponding p ocess {S( , s)} ≥s, whe e S( , s)x0 ep esen s he solu ion o he sys em a ime ha was a x0a ime s(x0∈R2) (c . Sell [24]). Fo non-au onomous sys ems he ini ial ime is as impo an as he inal ime: his is in con as o he au onomous case in which only he ime elapsed is ele an , i.e. S( , s) = S( −s, 0) o all ≥s. Thus o au onomous sys ems he beha iou o solu ions S( , s)x0 o → ∞ is he same as o s→ −∞. Howe e , his “pullback” beha iou (which o ms he basis o he heo y o a ac o s in non-au onomous sys ems, as de eloped by Cheban e al. [5], Kloeden and Schmal uss [12], Schmal uss [23], C auel e al. [10]) is dis inc om he o wa ds asymp o ics in he non-au onomous case. In his pape we comple ely desc ibe he asymp o ic beha iou o (1) bo h as → ∞ and as s→ −∞. The o wa ds asymp o ic beha iou is essen ially he same as ha o he au onomous sys em E[0], while he pullback asymp o ic beha iou is essen ially he same as ha o E[A]. Fo ce ain pa ame e anges hese beha iou s a e dis inc , †We no e he e ha while he heo y o asymp o ically au onomous equa ions (de eloped by Ma kus [17], Thieme [28], and Mischaikow e al. [18]) gi es some use ul in o ma ion abou he beha iou o he sys em as → ∞, a ull desc ip ion equi es he addi ional analysis ha we p esen he e. A non-au onomous Lo ka-Vol e a sys em 3 and he pullback p ocedu e p o ides mo e in o ma ion abou he sys em han would be a ailable solely by conside ing he o wa d asymp o ics. The main aim o his pape is no o o e a signi ican ad ance o he heo y o one pa icula class o equa ions (Lo ka-Vol e a models); a he , i is o illus a e he powe o he pullback idea in gi ing a ull desc ip ion o he dynamics, and o highligh some o he in e es ing p oblems ha can occu in non-au onomous sys ems. 1.1. P e ious esul s o non-au onomous sys ems The e a e a se ies o pape s ha ea non-au onomous N-dimensional Lo ka-Vol e a sys ems in which all he coe icien s a e allowed o be non-au onomous (see, among o he s, Ahmad [2], [3], Ahmad and Laze [4], Mon es de Oca and Zeeman [19], Redhe e [21]). We belie e ha we could also ea his mo e gene al p oblem, al hough we ha e chosen, o he sake o cla i y, o conside only he simples case whe e he p e ious me hods a e no di ec ly applicable. Fo (1) he condi ions in hese pape s become 0< a1≤a( )≤a2,(2) o all ∈R o he esul s in Ahmad [1] o [2], and bµ/c < λ < ·in ∈Ra( )¸µ/d (3) o apply hose in Ahmad and Laze [4] o Mon es de Oca and Zeeman [19] ( his la e in pa icula equi es ha a( )≥a > 0 as in Redhe e [21]). Unde hese hypo heses he abo e au ho s can p o e he exis ence o a s ic ly posi i e solu ion o he sys em ha is globally asymp o ically s able, while i λ < bµ/c o λ > ·sup ∈R a( )¸µ/d (4) hey ob ain ex inc ion o one o he species (i.e. he solu ion goes o ze o) as ime goes o in ini y. In ou case none o he condi ions (2–4) hold. 2. Pullback a ac o s in non-au onomous sys ems We now show how we can conside he solu ions o non-au onomous equa ions wi hin a dynamical sys ems-like amewo k, in oduce mo e ca e ully he pullback idea ha we make g ea use o , and show ha he equa ions in (1) gi e ise o an o de -p ese ing sys em. 2.1. P ocesses We deno e by S( , s)x he solu ion o (1) a ime ha akes he alue xa ime s. Then S( , s) de ines a p ocess, whe e a gene al p ocess on a comple e me ic space (X, d) is a amily o mappings {S( , s)} ≥s, , s ∈R ha sa is y: A non-au onomous Lo ka-Vol e a sys em 4 (i) S( , ) = IdX, o all ∈R, (ii) S( , τ)S(τ, s)u=S( , s)u o all s≤τ≤ , u ∈X, and (iii) u7→ S( , s)uis con inuous in X. 2.2. The pullback a ac o We now gi e a o mal de ini ion o he “pullback a ac ion” idea discussed in he in oduc ion. Fo A, B ⊂Xwe le dis (A, B) deno e he Hausdo semidis ance, dis (A, B) = sup a∈A in b∈Bd(a, b). De ini ion 2.1. A ime dependen amily {K( )} ∈Ris pullback a ac ing i o each ∈R lim s→−∞ dis (S( , s)D, K( )) = 0. o e e y bounded se D. This concep o a ac ion conside s a ixed inal ime and mo es he ini ial ime o −∞: his does no mean ha we a e going backwa ds in ime, bu a he ha we conside he s a e o he sys em a ime a ising om he same ini ial condi ion s a ing a ea lie and ea lie imes s. As ema ked abo e, in he au onomous case his no ion o pullback a ac ion ([12], [23]) is equi alen o he s anda d de ini ion. The gene alisa ion o he au onomous concep o in a iance is somewha mo e s aigh o wa d: De ini ion 2.2. A amily {B( )} ∈Ro subse s o Xis said o be in a ian wi h espec o he p ocess Si S( , s)B(s) = B( ) o all (s, )∈R2, s ≤ . The de ini ion o a non-au onomous a ac o combines hese no ions o a ac ion and in a iance. De ini ion 2.3. The amily o compac se s {A( )} ∈Ris said o be he global pullback a ac o associa ed o he p ocess Si i is in a ian , pullback a ac ing, and is minimal in he sense ha i {C( )} ∈Ris ano he amily o closed pullback a ac ing se s, hen A( )⊂C( ) o all ∈R. (Chepyzho and Vishik [6] de ine he concep o ke nel sec ions o non-au onomous dynamical sys ems: hese co espond o he ib es A( ) in he abo e de ini ion o a global pullback a ac o .) We now gi e a gene al esul simila o hose ha can be ound in C auel e al. [10] and Schmal uss [23]. The pa icula o m o he heo em gi en he e is modelled on a esul o andom a ac o s gi en in C auel [9]. Theo em 2.4. The e is a global pullback a ac o {A( )} ∈Ri and only i he e exis s a amily {K( )} ∈Ro compac pullback a ac ing se s. A non-au onomous Lo ka-Vol e a sys em 5 2.3. Comple e and hype bolic ajec o ies The simples example o an in a ian se is a ixed poin xwi h S( , s)x=x o all ≥s. Howe e , his is a e y s ong p ope y o equi e o a solu ion o a non-au onomous equa ion. One o he mos basic p oblems wi h he in es iga ion o non-au onomous sys ems is o ind a sensible gene alisa ion o he no ion o a ixed poin . A po en ial candida e is gi en by he no ion o a comple e ajec o y, a con inuous map :R→Xwi h S( , s) (s) = ( ), o all ≥s. This is simply a solu ion ( ) o he equa ion ha is de ined o all ∈R. Howe e , he e a e o cou se many such solu ions, and we would ideally like o pick ou ce ain “dis inguished” solu ions. In a p e ious pape (Langa e al. [14]) we highligh ed he impo ance o comple e ajec o ies wi h “ce ain well-de ined s abili y p ope ies” bu we e less han explici abou wha hese migh be. One ui ul concep is he no ion o a “hype bolic ajec o y” (see Malho a & Wiggins [16]) ha gene alises he idea o a hype bolic ixed poin . We will see in ou example ha al hough his concep is use ul i is no ideal wi hou some mino modi ica ions. Fo au onomous sys ems hype bolici y can be cha ac e ised using he eigen alues o he linea ised equa ion nea he ixed poin . In non-au onomous sys ems we need o in oduce he concep o an exponen ial dicho omy (see Coppel [8], Sacke & Sell [22]). De ini ion 2.5. Le A( )be a eal N×Nma ix and Φ( , s)be he undamen al N×N ma ix solu ion o dX/d =A( )X X(s) = I(5) so ha he solu ion o dξ/d =A( )ξwi h ξ(s) = ξ0is gi en by Φ( , s)ξ0. Then (5) has an exponen ial dicho omy i he e is a p ojec ion ope a o Pand cons an s Kand λ > 0 such ha kΦ( , s)Pk ≤ Ke−λ( −s) ≥s kΦ( , s)(I−P)k ≤ Keλ( −s) ≤s. (6) (This de ini ion can be gene alised by allowing he p ojec ion P o depend on in such a way ha i is in a ian , i.e. ha Φ( , s)P(s) = P( ) ≥s; see Siegmund [25] o de ails.) De ini ion 2.6. A comple e ajec o y x( )o dx/d = (x, )is said o be hype bolic i he linea ised equa ion dX/d = D (x( ), )X has an exponen ial dicho omy. A non-au onomous Lo ka-Vol e a sys em 6 As wi h he hype bolici y o ixed poin s, he hype bolici y o he ajec o y x( ) gi es ise o local s able and uns able mani olds which a e p ese ed unde pe u ba ion. We ep oduce he e om [16] only he esul s ha a e ele an in his pape : o mo e de ails and a p oo see Yi [29]. In he s a emen o he heo em Nδ(x( )) deno es he ubula neighbou hood o x( ) in he space RN×R, Nδ(x( )) = [ ∈R (B(x( ), δ), ) ( he no a ion B(x, δ) is he open ball in RNo adius δcen ed a x). Theo em 2.7. Le x( )be a hype bolic ajec o y in RNsuch ha he p ojec ion P om de ini ion 2.5 has ank k. Then he e exis s a k+ 1 dimensional C mani old Ws loc( ), an n−k+ 1 dimensional C mani old Wu loc( ), and a δsuch ha (i) Ws loc is posi i ely in a ian , while Wu loc is nega i ely in a ian , and he wo mani olds in e sec along x( ), (ii) ajec o ies on Ws loc con e ge owa ds x( )exponen ially as as ime uns o wa ds, and lea e Nδ(x( )) as ime uns backwa ds, while (iii) ajec o ies on Wu loc lea e Nδ(x( )) as ime uns o wa ds bu con e ge exponen ially as owa ds x( )as ime uns backwa ds, and (i ) any ajec o y s a ing in Nδ(x( )) ha is no in Ws loc no Wu loc will lea e Nδ(x( )) bo h as ime uns o wa ds and as ime uns backwa ds. ( ) Le dx/d = (x, , p)be a amily o non-au onomous ODEs ha a y wi h he pa ame e pin a C manne (uni o mly on se s o he o m K×Rwhe e Kis any compac subse o RN). Then any hype bolic ajec o y ha exis s o some alue p=p0pe sis s unde small pe u ba ions, as do i s s able and uns able mani olds: hese depend on pin a C ashion. We will wan o compa e he dynamics o ou asymp o ically au onomous equa ion wi h i s “limi equa ion” E[0]. We can ea his case in he con ex o he abo e heo em by de ining a C1 amily o unc ions αp( ) such ha αp( ) = (a(1/p) ≤1/p a( ) ≥(1/p)+1 and αp( ) is mono onic be ween = 1/p and = (1/p) + 1. Then he amily o non-au onomous sys ems E[αp( )] con e ge owa ds E[0] in he app op ia e ashion as p→0. Since he dynamics o he sys ems E[αp( )] a e he same as hose o E[a( )] o ≥(1/p) + 1 we expec ha pe u ba ions o hype bolic ajec o ies o E[0] will play a ole in unde s anding he dynamics o E[a( )]. 2.4. O de p ese ing sys ems When a non-au onomous sys em is “o de -p ese ing” (in a sense ha we now make p ecise) i is possible o ob ain mo e in o ma ion abou he s uc u e o i s pullback A non-au onomous Lo ka-Vol e a sys em 7 a ac o . This p og amme is ca ied ou in some de ail in Chuesho [7], Langa & Su´a ez [13], and Smi h [26]. 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)w1¹S( , s)w2 o all ≥s. Fo ou equa ions we can de ine an app op ia e ela ion ¹on R2 ha makes he sys em o de -p ese ing: gi en (u1, 1),(u2, 2) we say ha (u1, 1)¹(u2, 2)⇐⇒ u1≤u2and 1≥ 2.(7) In ac he ollowing sligh ly s onge esul (c . Hess and Laze [11]) will be use ul. We will deno e he p ocess co esponding o he sys em E[ ( )] by S ( )( , s), ese ing S( , s) o he p ocess co esponding o (1). Lemma 2.8. Le ( )and ˜ ( )be non-nega i e, and deno e by (u( ), ( )) he solu ion o E[ ( )] wi h (u(s), (s)) = (us, s)and by (˜u( ),˜ ( )) he solu ion o E[ ˜ ( )] ha sa is ies (˜u(s),˜ (s)) = (˜us,˜ s). Then, p o ided ha ( )≥˜ ( ) o all ∈R, (us, s)¹(˜us,˜ s) =⇒S ( )( , s)(us, s)¹S˜ ( )(˜us,˜ s) o all ≥s. We w i e “S ( )¹S˜ ( )”. P oo . Assume ini ially ha ( )>˜ ( ) o all ∈R. Le [s, T] be he maximal in e al on which (u( ), ( )) ¹(˜u( ),˜ ( )) o ∈[s, T] (8) and suppose ha T < ∞. A ime Tone o he ollowing h ee possibili ies occu s: (i) u(T) = ˜u(T) = ubu (T)>˜ (T): clea ly o some δ1>0 we ha e ( )≥˜ ( ) o ∈(T, T +δ1]. We also ha e d d (˜u−u)(T) = u[( (T)−˜ (T))u+b( (T)−˜ (T))].(9) Since (T)>˜ (T) his is s ic ly posi i e and so o some δ2>0 we ha e u( )<˜u( ) o ∈(T, T +δ2]. This gi es (8) on [s, T +δ], con adic ing he maximali y o T. (ii) (T) = ˜ (T) = bu u(T)<˜u(T): a simila a gumen can be used o show ha (8) holds on [s, T +δ] o some δ > 0, since we ha e d d (˜ − )(T) = d(u(T)−˜u(T)) <0.(10) (iii) (u(T), (T)) = (˜u(T),˜ (T)) = (u, ): when =Twe ha e ( om (9) and (10)) d d (˜u−u)(T) = u( (T)−˜ (T)) and d d (˜ − )(T) = 0. Since (T)>˜ (T) we ha e ˜u( )> u( ) on (T, T +δ1] o some δ1>0. We also ha e d2 d 2(˜ − )(T) = − d d d (˜u−u)(T)<0, which implies ha ˜ ( )< ( ) on (T, T +δ2] o some δ2>0. Once again his gi es (8) on he longe in e al [s, T +δ] (wi h δ= min(δ1, δ2)). A non-au onomous Lo ka-Vol e a sys em 8 This p o es he esul o ( )>˜ ( ). I ( )≥˜ ( ) hen o each ² > 0 apply he abo e esul wi h ( ) eplaced by ( ) + ²; no ing ha he solu ions o E[ ( )] depend con inuously on ( ) he esul ollows as s a ed by aking he limi as ²→0. 2.5. The sepa a ix o an au onomous sys em In one o he p oo s below we will need o use p ope ies o he sepa a ix o he au onomous sys em (˙u=u(λ−au −b )u(0) = u0>0 ˙ = (µ−c −du) (0) = 0>0,(11) whe e ac < bd and aµ/d < λ < bµ/c. In his case he e a e h ee ixed poin s: wo s able ixed poin s a (0, µ/c) and (λ/a, 0), and one saddle poin in he in e io a x∗ a=1 bd −ac(bµ −cλ, dλ −aµ). We include a p oo since al hough he sys em is a s anda d example in unde g adua e di e en ial equa ions cou ses, we we e unable o ind any igo ous ea men in he li e a u e. In he p oo we use (x, y)≥(u, ) o mean x≥uand y≥ . P oposi ion 2.9. I ac < bd and aµ/d < λ < bµ/c hen he e is a sepa a ix Γa, gi en as he g aph o a s ic ly inc easing con inuous unc ion φa: (0,∞)→(0,∞)wi h lim u→0φa(u) = 0 such ha i 0     < = >     φa(u0) hen lim →∞ S( , 0)(u0, 0) =      (λ/a, 0) x∗ a (0, µ/c)     . Fu he mo e o each ixed u he alue o φa(u)is con inuous in aand is mono onically dec easing in a. P oo . Fi s we conside he p oblem o a ixed alue o a. Dulac’s c i e ion (∇ · [1 u (u, )] <0 in P, whe e is he igh -hand side o (11)) shows ha he equa ion has no pe iodic o bi s in P, and a simple compu a ion o he ime de i a i e o µ|u|2+λ| |2shows ha all o bi s a e bounded. I ollows om he Poinca ´e-Bendixson heo em ha e e y o bi con e ges o one o he ixed poin s. Le Γabe he s able mani old o x∗ a, i.e. Γa={x:Sa( , 0)x→x∗ aas →+∞}. This is he global ex ension o he local s able mani old o x∗ a, which is angen o he linea s able mani old a x∗ a; elemen a y conside a ions o he linea isa ion show ha he linea s able mani old mo es in o (u, )> x∗ aand (u, )< x∗ a. Since ˙uand ˙ a e A non-au onomous Lo ka-Vol e a sys em 9 bo h posi i e [nega i e] when (u, )> x∗ a[(u, )< x∗ a] i ollows ha close o x∗ a he sepa a ix Γais he g aph o a s ic ly inc easing con inuous unc ion φa; he con inui y o φain awi hin his neighbou hood is a s anda d esul om he heo y o local s able mani olds. The global s able mani old consis s o wo ajec o ies. In o de o unde s and hei beha iou ou side a neighbou hood o x∗ awe conside he ime- e e sed low, w i ing ˇu( ) o u(− ). Then since ˙ ˇuand ˙ ˇ can be bounded below o (u, )≥x∗ a+ (², ²), and i ˇ ( )→ ∞ hen we mus also ha e ˇu( )→ ∞ (i ˇu( )→c < ∞ hen ˙ ˇu→ ∞, a con adic ion) he s able mani old Γaex ends o in ini y in such a way ha φais de ined o e e y u > [x∗ a]1. Since (ˇu, ˇ )< x∗ ais in a ian o he ime e e sed low and his egion con ains no pe iodic o bi s (Dulac’s c i e ion again) i ollows ha he ajec o y o he le o x∗ acon e ges o he o igin as → −∞. The con e gence o S( , 0)(u0, 0) o one o he ixed poin s on he axes when he ini ial condi ion lies abo e/below Γais now immedia e, since Γais in a ian and consis s o all poin s a ac ed o x∗ a. The con inui y o φa(u) o all alues o uis a consequence o he con inuous dependence o solu ions on ini ial condi ions and on he pa ame e a, and i only emains o show ha φa(u) is mono onically dec easing. This will ollow i we can show ha o dis inc alues o a6= ˜a he sepa a ices Γaand Γ˜aa e disjoin , since x∗ a∈Γais s ic ly inc easing wi h espec o he o de ¹( his ollows om a simple calcula ion o dx∗ a/da). Wi hou loss o gene ali y assume ha a > ˜a, and suppose ha x∈Γa∩Γ˜a. Using lemma 2.8 we ha e Sa¹S˜a, and so x∗ a= lim →∞ Sa( , 0)x¹lim →∞ S˜a( , 0)x=x∗ ˜a. Howe e , x∗ aÂx∗ ˜a, a con adic ion. So Γa∩Γ˜a=∅as equi ed. 2.6. A non-au onomous logis ic equa ion Also use ul will be he ollowing simple esul ha gi es some p ope ies o he solu ions o he non-au onomous logis ic equa ion dx/d =x(p( )−l( )x)x(s) = x0,(12) wi h p( )>0 and l∈C0(R) wi h l( )>0 o all ∈R. We deno e he solu ion o his equa ion as θ[p(·),l(·)]( , s;x0), and no e ha i can be gi en explici ly by θ[p(·),l(·)]( , s;x0) = eR sp(u) du x−1 0+R seR sp(u) dul( ) d .(13) F om he e i is easy o deduce he ollowing p ope ies: Lemma 2.10. The solu ions o (12) ha e he ollowing p ope ies: (i) I p( )→p > 0and l( )→0as → ∞ hen θ[p,l]( , s)→ ∞ as → ∞. A non-au onomous Lo ka-Vol e a sys em 16 4.2.2. A pullback a ac ing coexis en s a e. Finally we in es iga e he pa ame e ange in which we ob ain ou pullback a ac ing coexis en s a e: he au onomous sys em E[A] has an a ac ing in e io ixed poin o his se o pa ame e s. In wo p oo s o his sec ion we will make use o he ollowing esul om Ahmad and Laze [4] (Lemma 3), ew i en he e using ou o de no a ion. Lemma 4.4. Suppose ha he e exis δ,δ1, δ2>0such ha o some ixed ∈R δ1a( )> δ2d+δ(20) δ2c > δ1b+δ, and ha he e exis solu ions x1(·)and x2(·)o (1) such ha a−¹xi(s)¹a+ o all s≤ whe e a−¹a+and bo h a e elemen s o P. Then x1( ) = x2( ) o all ∈R. Theo em 4.5. I Ac > bd and bµ/c < λ < Aµ/d (21) hen he e exis s a comple e ajec o y (U( ), V ( )) ∈Psuch ha o each u0, 0>0 and e e y ∈R, lim s→−∞ S( , s)(u0, 0) = (U( ), V ( )). P oo . Fo any ² > 0 such ha (A−²)c > bd he e exis s a 0(²) such ha a( )>(A−²) o all ≤ 0. I ollows ha o s, ≤ 0we ha e SA( , s)¹S( , s)¹SA−²( , s). E e y sys em E[a] wi h A−²≤a≤Ahas an a ac ing in e io ixed poin x∗ a, and o he pa ame e ange conside ed he e x∗ ais dec easing (wi h espec o he o de ¹) in a. I ollows ha x∗ A¹lim s→−∞ S( , s)(u0, 0)¹x∗ A−². Since S( 0, )xdepends con inuously on x, i ollows ha o any 0 S( 0, )x∗ A¹lim s→−∞ S( 0, s)(u0, 0)¹S( 0, )x∗ A−² and heo em 2.4 ensu es he exis ence o a non-au onomous a ac o A( ). Any wo ajec o ies x1( ) and x2( ) in A( ) mus sa is y x∗ A¹xi( )¹x∗ A−² o ≤ 0. Lemma 4.4 now gua an ees ha x1( ) = x2( ) o all ∈R, and hus A( ) consis s o a single ajec o y (U( ), V ( )) as claimed. A non-au onomous Lo ka-Vol e a sys em 17 5. Conclusions We ha e desc ibed in some de ail he dynamics o a wo-dimensional non-au onomous compe i i e Lo ka-Vol e a model ha is asymp o ically au onomous bo h as → ∞ and as → −∞. While he asymp o ic beha iou as → ∞ co esponds o ha o he limi ing sys em a in he u u e, he pullback asymp o ic beha iou as s→ −∞ appea s o co espond o ha o he limi ing sys em in he dis an pas . We hink ha a he e y leas his example should se e o cla i y he ype o in o ma ion ha can be picked up using he pullback idea. I i eally is o wa d asymp o ic beha iou ha is o in e es hen we can expec o gain li le om he pullback app oach, bu i a p opensi y o a ou his poin o iew is a p oduc only o i s amilia i y (and he equi alence o he wo no ions in he au onomous case) hen he pullback p ocedu e p o ides ano he echnique ha can be use ul in unco e ing impo an quali a i e ea u es o he dynamics. Fo example, o a ce ain ange o pa ame e s he e is a dis inguished posi i e “coexis en ” ajec o y x( ) = (u( ), ( )) ha is pullback a ac ing. To gi e his a biological in e p e a ion, i we e o a i e oday ( = 0) a a emo e island on which wo species ha e been compe ing acco ding o (1) o a long ime, he dis ibu ion o he wo species would be e y close o (u( ), ( )). Howe e , we know ha one o he species is des ined o die ou in he u u e. Some in e es ing ma hema ical ques ions a e also aised. When Ais su icien ly small he sys em E[a( )] will be C1close o E[0] o e he whole line ( ∈R) and he saddle poin x∗o he sys em E[0] will become a hype bolic ajec o y o E[a( )]: ou esul s con i m his, so ha when Ac < bd he pullback beha iou and he o wa ds asymp o ic beha iou a e simila . Howe e , when Ac > bd he pic u e is di e en : somehow we ha e o “join” he pullback beha iou (an a ac ing coexis en ajec o y) o he o wa ds beha iou : i is no clea ha he hype bolic ajec o y emana ing om he s able posi i e ixed poin o E[A] emains hype bolic o all ∈R. In pa icula i seems mo e na u al o allow o “e en ually hype bolic” ajec o ies whe e we only equi e an exponen ially dicho omy o , s ≥To , s ≤T( o some app op ia e T). Wi h u he analysis we belie e ha i would ha e been possible o ea no only he p eda o -p ey and coope a i e cases, bu also highe -dimensional sys ems and mo e gene al non-au onomous e ms (indeed, a ela ed in ini e-dimensional p oblem is s udied in Langa e al. [15]). 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