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]). Howe e , we ha e p e e ed o keep he p oblem ela i ely simple
in o de o show ha he pullback p ocedu e can be a e y use ul ool.
Acknowledgmen s
This wo k has been pa ially suppo ed by P oyec o D.G.I.C.Y.T. (Spain) BFM2002-
03068. JCR is a Royal Socie y Uni e si y Resea ch Fellow, and would like o hank he
Socie y o all hei suppo . He would also like o hank EDAN o hei hospi ali y,
and Ibe d ola o hei gene osi y.
A non-au onomous Lo ka-Vol e a sys em 18
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