ON THE SYMBIOTIC LOTKA-VOLTERRA MODEL
WITH DIFFUSION AND TRANSPORT EFFECTS
M. Delgado1, J. L´
opez-G´
omez2and A. Su´
a ez1
1Dp o. de Ecuaciones Di e enciales y An´alisis Num´e ico
Uni e sidad de Se illa
C. Ta ia s/n. 41012-Se illa, Spain
2Depa amen o de Ma em´a ica Aplicada
Uni e sidad Complu ense
28040-MADRID, Spain
Abs ac . In his wo k we analyze he exis ence, s abili y and mul iplici y o coexis ence
s a es o a symbio ic Lo ka-Vol e a model wi h gene al di usi i ies and anspo e ec s.
Global bi u ca ion heo y, blowing up a gumen s o a p io i bounds, singula pe u ba ion
esul s, singula i y heo y and ixed poin index in cones a e among he echniques used
o ge ou esul s and o explain he d as ic change o beha io exhibi ed by he dynamics
o he model be ween he cases o weak and s ong mu ualism be ween he species. Ou
me hodology wo ks ou o ea much mo e gene al classes o symbio ic models.
AMS Subjec Classi ica ion: 35K57, 35B25, 35B32, 35B45, 35B50.
Key wo ds and ph ases: Blowing up o a p io i bounds in sys ems. Local and global
bi u ca ion heo y. Singula i y heo y. Fixed poin index in cones. Singula pe u ba-
ions.
1. In oduc ion. In his pape we analyze he exis ence, mul iplici y and s abili y o
coexis ence s a es o he ollowing p oblem
L1u=λu −a(x)u2+b(x)u
L2 =µ −d(x) 2+c(x)u in Ω ,(1.1a)
u= = 0 on ∂Ω,(1.1b)
Typese by A
M
S-T
EX
Typese by A
M
S-T
EX
1
2 M. DELGADO, J. L´
OPEZ-G´
OMEZ AND A. SU´
AREZ
whe e Ω is a bounded domain o RNwi h bounda y ∂Ω o class C2 egula i y, Lk,
k= 1 ,2 a e wo second o de uni o mly ellip ic ope a o s o he o m
Lk=−
N
X
i,j=1
aijk(x)∂i∂j+
N
X
j=1
bjk(x)∂j+ck(x)k= 1 ,2,(1.2)
wi h
aijk ∈C(Ω) , bjk , ck∈L∞(Ω) , i , j ∈ {1, ..., N}, k ∈ {1,2},(1.3)
and a,b,c,d∈C(Ω) sa is y a(x)>0, d(x)>0, o each x∈Ω, and b≥0, c≥0 in Ω,
b6= 0, c6= 0; λ,µ∈Rwill be ega ded as bi u ca ion pa ame e s. Unde hese assump-
ions, (1.1) p o ides us wi h a model o symbio ic species, whe e Ω is he inhabi ing
egion, u(x) and (x) a e he densi ies o each o he species, a(x) and d(x) desc ibe he
limi ing e ec s o c owding in each popula ion, b(x) and c(x) a e he in e ac ion a es
be ween he species, he ope a o s Lk−ck(x), k= 1,2, measu e he di usi i ies and
he ex e nal anspo e ec s o he species, and λ−c1(x), µ−c2(x) a e he g ow h
a es o he species, posi i e on a o able egions and nega i e on un a o able ones. In
his model we a e assuming ha Ω is ully 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.
In his wo k ou a en ion will be ocused in o he p oblem o analyzing he exis ence,
s abili y and mul iplici y o he non-nega i e solu ion couples (u, ) o (1.1). Due o he
s uc u e o (1.1) and hanks o he s ong maximum p inciple, i (u, ) is a solu ion
o (1.1) wi h u6= 0 ( esp. 6= 0), hen u( esp. ) is s ongly posi i e in he sense o
Sec ion 2. The e o e, (1.1) admi s h ee ypes o non-nega i e componen -wise solu ion
couples. Namely, he i ial one, (0,0); hose wi h one componen posi i e and he
o he ze o, (u, 0) o (0, ), e e ed as he semi- i ial posi i e solu ions, and hose wi h
bo h componen s posi i e, he coexis ence s a es.
The symbio ic model has a ac ed much less a en ion in he li e a u e han i s
compe ing and p eda o -p ey coun e pa s, due basically o he absence o a p io i
bounds o he coexis ence s a es in high spa ial dimensions (N≥6) unde s ong
mu ualism (bc −ad la ge). This lack o a p io i bounds was obse ed o iginally in [16],
whe e i was shown ha he posi i e solu ions o he pa abolic p oblem associa ed wi h
(1.1) may blow up in ini e ime when L1=L2=−∆ and bc > ad, and in [21], whe e
i was shown ha i in addi ion λ=µ, hen he coexis ence s a es o (1.1) a e gi en by
he posi i e solu ions o
−∆w=λw +w2in Ω , w|∂Ω= 0 ,(1.4)
and ha hanks o he esul s o [13], (1.4) possesses uni o m a p io i bounds in any
compac subin e al o λi , and only i , 2 <N+2
N−2, i.e. i N≤5.
The absence o a p io i bounds o he coexis ence s a es o (1.1) makes e y in ol ed
he p oblem o inding ou global su icien condi ions o he exis ence o a coexis ence
SYMBIOTIC SPECIES 3
s a e, since mos o he echnical ools a ailable o a ack his kind o p oblems in ol e
ei he deg ee heo y, i.e. global bi u ca ion heo y, o mono onici y echniques, whe e
he exis ence o a p io i bounds is needed. Ne e heless, al hough mos o he a en ion
has been ocused in o he e y special case when L1=L2=−∆ and a,b,c,da e
cons an s, in ecen yea s some subs an ial p og ess has been ca ied ou in o he analysis
o hese p oblems.
The s udy o symbio ic species ac ually s a ed in [19], whe e i was cons uc ed
mono onic sequences which app oxima e he solu ions o (1.1). In [17] he me hod
o sub and supe solu ions o sys ems, coming om [28], was used o show ha i
λ > σ1and µ>σ1, hen (1.1) possesses a coexis ence s a e i , and only i , bc < ad,
whe e σ1is he p incipal eigen alue o −∆ in Ω unde homogeneous Di ichle bounda y
condi ions. This esul was gene alized in [20] o co e some mo e gene al classes o
symbio ic kine ics. The i s global esul abou he exis ence o coexis ence s a es
o he symbio ic model was ound in [25] by using global bi u ca ion heo y, whe e i
was shown ha i any o he semi- i ial posi i e solu ions is linea ly uns able, hen
he model possesses a coexis ence s a e p o ided bc < ad; global in he sense ha i
some o he semi- i ial s a es is s able, hen he e a e choices o he se e al pa ame e s
in ol ed in he se ing o (1.1) o which he model does no admi a coexis ence s a e
(c . Sec ion 11 he e in o u he de ails). Almos simul aneously, in [35] was ound he
same esul included in [25], bu his ime using he me hod o sub and supe solu ions.
Mo e ecen ly, allowing he coe icien s o he model o a y, he echnique o decoupling
was shown o wo k ou o ge he same esul as in [25] and [35], [5]. In [21] and [23]
ixed poin index in cones and global bi u ca ion heo y we e shown o wo k ou o ge
he co esponding esul s o wide classes o models.
Al hough he global esul s o [21] wo k ou o show ha a global con inuum o
coexis ence s a es emana es om each o he su aces o semi- i ial posi i e solu ions
along hei cu es o change o s abili y in he space o he pa ame e s (λ, µ), he i s
global esul in he case bc > ad was ound in [27], whe e i was shown ha i N≤5 and
some o he semi- i ial posi i e solu ions is linea ly s able, hen he model possesses a
coexis ence s a e. We poin ou ha his esul was ob ained o he special case when
L1=L2=−∆ and all coe icien s a e cons an . In [27], he blowing up a gumen o
[13] was adap ed o show he exis ence o a p io i bounds in case N≤5 and hen he
ixed poin index in cones was used o comple e he p oo .
In his wo k we ex end and comple e all he p e ious ea u es, ob aining in addi ion
some op imal non-exis ence and mul iplici y esul s o all anges o he pa ame e s in
he gene al se ing o (1.1), and in addi ion we analyze he bi u ca ion equa ions o (1.1)
a (λ, µ) = (σΩ
1[L1], σΩ
1[L2]). He ea e , gi en an ellip ic ope a o L,σΩ
1[L] will s and o
he p incipal eigen alue o Lin Ω unde homogeneous Di ichle bounda y condi ions.
Ou analysis o he bi u ca ion equa ions a (σΩ
1[L1], σΩ
1[L2]) explains he d as ic change
o beha io o he global con inuum o coexis ence s a es as some o he in e ac ions
be ween he species, bo c, g ows ac ossing he c i ical alue gi en by Theo em 10.1 in
Sec ion 10. Namely, he global mani old o coexis ence s a es linking he wo su aces o
semi- i ial posi i e solu ions u ns backwa ds in he pa ame e space (λ, µ) changing
4 M. DELGADO, J. L´
OPEZ-G´
OMEZ AND A. SU´
AREZ
i s ela i e posi ion wi h espec o each he su aces o semi- i ial solu ions, as he
ampli ude o b, o c, g ows.
To s a e ou main esul s, we ha e o in oduce some o no a ion. Gi en λ > σΩ
1[L1]
( esp. µ > σΩ
1[L2]), (θλ,0) ( esp. (0, θµ)) will s and o he unique semi- i ial solu ion
o (1.1) o he o m (u, 0), u > 0 ( esp. (0, ), > 0). Mo eo e , o any ∈L∞(Ω) we
deno e
L:= ess in
Ω , M:= ess sup
Ω
.
Among ou main esul s we lis he ollowing ones:
•I bMcM< aLdLand any o he semi i ial posi i e solu ions is linea ly uns able,
hen (1.1) possesses a coexis ence s a e. I in addi ion λ > σΩ
1[L1] and µ > σΩ
1[L2], hen
he e exis s I0>0 such ha i
min {bM, cM}< I0,
hen he coexis ence s a e is unique and exponen ially asymp o ically s able.
•I bMcM< aLdLand o (λ, µ) = (λ0, µ0) some o he semi i ial posi i e solu ions
is linea ly s able and (1.1) possesses a coexis ence s a e, hen i possesses a coexis ence
s a e o each (λ, µ) sa is ying λ≥λ0,µ≥µ0, and a leas wo coexis ence s a es i
λ > λ0,µ > µ0and some o he semi- i ial posi i e solu ions is linea ly s able.
•I bMcM< aLdL, hen o each λ∈R, he e exis s µex (λ)∈Rsuch ha (1.1)
does no admi a coexis ence s a e i µ≤µex (λ). Simila ly, o each µ∈R, he e exis s
λex (µ)∈Rsuch ha (1.1) does no admi a coexis ence s a e i λ≤λex (µ).
•I L1=L2,N≤5,
bLcL−aMdM>max {aMbM−aLbL, dMcM−dLcL},(1.5)
and some o he semi i ial posi i e solu ions is linea ly s able, hen (1.1) possesses a
coexis ence s a e.
•Assume ha L1=L2,N≤5, (1.5), and ha he e exis s (λ, µ)=(λ0, µ0) o
which (1.1) possesses a coexis ence s a e being any o he semi- i ial s a es linea ly
uns able. Then, (1.1) possesses a coexis ence s a e o each (λ, µ) sa is ying λ≤λ0and
µ≤µ0, and a leas wo coexis ence s a es i λ < λ0,µ < µ0and any o he semi- i ial
s a es is linea ly uns able.
•Assume L1=L2,N≤5 and (1.5). Then, o each λ∈R he e exis s µex (λ)∈R
such ha (1.1) does no admi a coexis ence s a e i µ≥µex (λ). Simila ly, o each
µ∈R, he e exis s λex (µ)∈Rsuch ha (1.1) does no admi a coexis ence s a e i
λ≥λex (µ).
We now desc ibe he dis ibu ion and con ains o his pape . In Sec ion 2 we gi e an
ex ension o Theo em 2.5 in [22] o co e ou gene al se ing he e in, and hen use i o
in e some basic mono onici y p ope ies o p incipal eigen alues. Mos o hese esul s
come om Sec ion 2 o [3].
SYMBIOTIC SPECIES 5
In Sec ion 3 we s udy he single bounda y alue p oblem
L1u=λ u −a(x)u2in Ω , u|∂Ω= 0 .(1.6)
A pa icula a en ion is paid o he beha io o i s unique posi i e solu ion as λ↑ ∞,
showing ha
lim
λ↑∞
θ[L1,λ,a]
λ=a−1(1.7)
uni o mly on any compac subse o Ω, whe e θ[L1,λ,a]s ands o he unique posi i e
solu ion o (1.6). This esul ex ends he co esponding singula pe u ba ion esul in
Sec ion 3 o [12] o ou gene al se ing he e in, and i is he basic echnical ool o ge
ou non-exis ence esul s in Sec ion 7.
In Sec ion 4 we cha ac e ize he a ac i e cha ac e o each o he semi i ial posi i e
solu ions in e ms o se e al pa ame e s in ol ed in he se ing o (1.1) h ough by he
p incipal eigen alues o some ela ed second o de ellip ic ope a o s. Then, we analyze
he shape o he cu es o change o s abili y in he space o he pa ame e s (λ, µ).
Sec ion 5 is de o ed o he abs ac esul s conce ning he exis ence o global con inua
o coexis ence s a es emana ing om he su aces o semi i ial posi i e solu ions along
hei espec i e cu es o change o s abili y. The analysis h oughou his wo k shows
ha hese esul s a e op imal, educing he p oblem o inding ou coexis ence s a es
o (1.1) o he p oblem o inding ou a p io i bounds o he componen -wise posi i e
solu ions o (1.1). The me hodology adop ed in his sec ion comes om he abs ac
heo y de eloped in [21] o gene al sys ems wi h wo species.
In Sec ion 6 we analyze he exis ence o coexis ence s a es o he case o small
in e ac ion coe icien s. How small should hey a e is measu ed by condi ion
bMcM< cLdL.(1.8)
P ecisely, we will ind ou some non-exis ence esul s and hen we will use he heo y
o Sec ion 5 o show ha (1.1) possesses a coexis ence s a e i any o he semi i ial
posi i e solu ions is linea ly uns able. The analysis o Sec ion 11 o he case o cons an
coe icien s will show he op imali y o ou esul s.
In Sec ion 7 we analyze he exis ence o coexis ence s a es o he case o la ge
in e ac ion coe icien s. How la ge should hey a e is measu ed by condi ion (1.5).
No ice ha i any coe icien is assumed o be cons an , hen (1.5) becomes in o
bc > ad . (1.9)
By echnical easons o mos o he esul s in his sec ion we need assuming ha
L1=L2, assump ion needed in all p e ious e e ences. We begin he sec ion gi ing a
necessa y condi ion o he exis ence o coexis ence s a es which is o ally new e en o
he simples symbio ic models whe e L1=L2=−∆ and any coe icien is cons an .
Namely, i (0, θ[L1,µ,d]) ( esp. (θ[L1,λ,a])) is linea ly uns able, hen (1.1) does no admi a

6 M. DELGADO, J. L´
OPEZ-G´
OMEZ AND A. SU´
AREZ
coexis ence s a e i µ( esp. λ) is su icien ly la ge (c . Theo em 7.1 he e in). This non-
exis ence esul is based upon (1.7), inding ou he beha io o an e en ual sequence o
coexis ence s a es o µ, o λ, la ge. Then, we adap he blowing up a gumen o [13]
o show ha uni o m a p io i bounds o he coexis ence s a es o (1.1) a e a ailable i
N≤5. We should poin ou ha ou blowing up a gumen di e s subs an ially om
he co esponding a gumen o [27] and ha we need a gene al Liou ille ype esul
much sha pe han he co esponding esul in [27]. These addi ional di icul ies coming
om he ac ha in his wo k we a e dealing wi h a gene al ellip ic ope a o and wi h
spa ially a ying coe icien s. We e e o Sec ion 7 o u he de ails. B inging oge he
he non-exis ence esul s and he a p io i bounds, i ollows om he global esul s in
Sec ion 5 ha i any o he semi i ial posi i e solu ions is linea ly s able, hen (1.1)
possesses a coexis ence s a e.
In Sec ion 8 we use he abs ac heo y o [2] o show ha he me hod o sub and
supe solu ions is alid o (1.1). Then, we use i o analyze he s uc u e o he se
o λ’s and µ’s o which (1.1) possesses a coexis ence s a e and o ge ou mul iplici y
esul s, hose al eady s a ed in he lis abo e.
In Sec ion 9 we ob ain simple eadily compu able condi ions in e ms o he se e al
coe icien s in ol ed in he se ing o (1.1) ensu ing ha (1.1) has a unique s able co-
exis ence s a e, and hen conside he pa abolic p oblem associa ed wi h (1.1) o show
ha he e is a dense subse o he se o ini ial da a such ha any solu ion s a ing
he e in con e ges o he coexis ence s a e as ime g ows o in ini y.
In Sec ion 10, conside ing (λ, µ) as he main bi u ca ion pa ame e s we desc ibe he
possible local bi u ca ion diag ams nea he co-dimension wo singula i y
(λ, µ) = (σΩ
1[L1], σΩ
1[L2]) .
Fo his, we apply he gene al esul s o [10] whe e one o he au ho s de eloped a
singula i y heo y o deal wi h his ype o wo pa ame e bi u ca ion p oblems.
Finally, in Sec ion 11 we es ic ou sel es o he o iginal Lo ka-Vol e a symbio ic
model wi h di usion, L1=L2=−∆ and a,b,c,dcons an s, o which we can gi e
some sha pe exis ence and non-exis ence esul s and can go u he in he analysis o
he bi u ca ion equa ion a ound he co-dimension wo bi u ca ion poin , ob aining in
addi ion some global esul s abou he na u e o he local bi u ca ions o coexis ence
s a es om he su aces o semi i ial posi i e solu ions along hei cu es o change o
s abili y. As a esul om his analysis we can explain he d as ic change o beha io
o he global mani old o coexis ence ha links he wo su aces o semi i ial posi i e
solu ions along hei cu es o change o s abili y as bc ac osses he c i ical alue ad
passing om alues whe e bc < ad o alues whe e bc > ad.
2. The maximum p inciple. Main p ope ies o he p incipal eigen alues. In
his sec ion we gi e an ex ension o Theo em 2.5 in [22] o co e ou se ing he e and hen
we in e some basic p ope ies o p incipal eigen alues which will be used h oughou
SYMBIOTIC SPECIES 7
his pape . We will conside a uni o mly ellip ic ope a o o he o m
L=−
N
X
i,j=1
aij(x)∂i∂j+
N
X
j=1
bj(x)∂j+e(x),(2.1)
wi h
aij ∈C(Ω) , bj, e ∈L∞(Ω) , i , j ∈ {1, ..., N},(2.2)
and use he na u al p oduc o de on Lp(Ω) ×Lp(∂Ω). Recall ha p > N implies
W2,p(Ω) ⊂C2−N
p−ε(Ω) wi h compac imbedding o all ε > 0 and ha each u∈
W2,p(Ω) is a.e. wice classically di e en iable in Ω (e.g. Theo em VIII.1 o [33]).
Suppose ha p>N. Then u∈W2,p(Ω) is said o be s ongly posi i e i u(x)>0
o x∈Ω and ∂nu(x)<0 o all x∈∂Ω wi h u(x) = 0, whe e nis he ou wa d uni
no mal on ∂Ω. The ope a o Lis said o sa is y he s ong maximum p inciple in Ω i
p > N,u∈W2,p(Ω), and (Lu, u)>(0,0) imply ha uis s ongly posi i e. Conside
he eigen alue p oblem
Lu=σu in Ω , u = 0 on ∂Ω,(2.3)
in W2,p(Ω) and le Lpdeno e he closu e o he ope a o L|W2,p(Ω)∩W1,p
0(Ω) in Lp(Ω).
Then, (2.3) can be e o mula ed as he eigen alue equa ion
Lpu=σu in Lp(Ω) .(2.4)
I is an easy consequence o s anda d egula i y heo y ha he spec um and he
eigenspaces o Lpa e independen o p > N. Mo eo e , om he s ong maximum
p inciple and he gene aliza ion o he K ein Ru man Theo em o [32] oge he wi h
Theo em 3 in [29], he ollowing esul holds (c . Sec ion 2 o [3]).
Theo em 2.1. The e exis s a leas eigen alue o (2.4), deno ed by σΩ
1[L]and called
p incipal eigen alue o Lin Ω. This eigen alue is simple and possesses a unique eigen-
unc ion, up o mul iplica i e cons an s, which can be aken posi i e, he so called p in-
cipal eigen unc ion o Lin Ω. Mo eo e , he p incipal eigen unc ion is s ongly posi i e
and σΩ
1[L]is he only eigen alue o (2.4) possessing a posi i e eigen unc ion. Fu he -
mo e, any o he eigen alue σo (2.4) sa is ies
Re σ > σΩ
1[L]
and (Lp+ν)−1∈ L(Lp(Ω)) is posi i e, compac and i educible o ν > −σΩ
1[L].
I p > N a unc ion u∈W2,p(Ω) is said o be a posi i e supe solu ion o Lin Ω i
u≥0 and (Lu, u)≥(0,0). I in addi ion (Lu, u)>(0,0), hen i is said ha uis a
posi i e s ic supe solu ion. Simila ly, a unc ion u∈W2,p(Ω) is said o be a posi i e
subsolu ion o Lin Ω i u≥0 and (Lu, u)≤(0,0). I in addi ion (Lu, u)<(0,0), hen
i is said ha uis a posi i e s ic subsolu ion.
¿F om he s ong maximum p inciple i is easily seen ha any posi i e s ic su-
pe solu ion is s ongly posi i e. Mo eo e , he ollowing cha ac e iza ion o he s ong
maximum p inciple holds (c . Theo em 2.5 in [22] and Theo em 2.4 in [3]).
8 M. DELGADO, J. L´
OPEZ-G´
OMEZ AND A. SU´
AREZ
Theo em 2.2. The ollowing asse ions a e equi alen :
(i) σΩ
1[L]>0;
(ii) Lpossesses a posi i e s ic supe solu ion in Ω;
(iii) Lsa is ies he s ong maximum p inciple in Ω.
F om his cha ac e iza ion we can eadily ge he ollowing p ope ies o σΩ
1[L] which
will be used h oughou his wo k. Fo sel adjoin ope a o s, hese p ope ies a e easily
ob ained om he a ia ional cha ac e iza ion o he p incipal eigen alue.
Theo em 2.3. (i) Mono onici y wi h espec o he po en ial: Le V1,V2∈L∞(Ω) such
ha V1≤V2and V1< V2on a se o posi i e measu e. Then,
σΩ
1[L+V1]< σΩ
1[L+V2].(2.5)
(ii) Con inui y wi h espec o he po en ial: I Vn∈L∞(Ω),n≥1is a sequence o
po en ials such ha
lim
n→∞ kVn−Vk∞,Ω= 0 ,
hen
lim
n→∞ σΩ
1[L+Vn] = σΩ
1[L+V].
(iii) I Ω1is a p ope subdomain o Ωwi h ∂Ω1o class C2, hen
σΩ1
1[L]> σΩ
1[L].(2.6)
P oo . (i) Le ϕ1be he p incipal eigen unc ion associa ed wi h σΩ
1[L+V1]. Then,
(L+V2)ϕ1=σΩ
1[L+V1]ϕ1+ (V2−V1)ϕ1> σΩ
1[L+V1]ϕ1
on a se o posi i e measu e, and hence ϕ1is a posi i e s ic supe solu ion o L+V2−
σΩ
1[L+V1]. Thus, hanks o Theo em 2.2, we ind ha
σΩ
1[L+V2−σΩ
1[L+V1]] >0.
This ela ion implies (2.5).
(ii) Fo any ε > 0 he e exis s N0∈Nsuch ha
V−ε≤Vn≤V+ε∀n≥N0.
Thus, by Pa (i) we ind ha
σΩ
1[L+V]−ε≤σΩ
1[L+Vn]≤σΩ
1[L+V] + ε .
This comple es he p oo .
(iii) Le ϕdeno e he p incipal eigen unc ion associa ed wi h σΩ
1[L]. Then,
(L−σΩ
1[L])ϕ= 0
in Ω1and ϕ > 0 on ∂Ω1. Thus, ϕis a posi i e s ic supe solu ion o L−σΩ
1[L] in Ω1
and hence, i ollows om Theo em 2.2 ha
σΩ1
1[L−σΩ
1[L]] >0.
This ela ion implies (2.6). ¤
SYMBIOTIC SPECIES 9
3. The logis ic equa ion. The semi- i ial posi i e solu ions o (1.1) a e gi en by
he posi i e solu ions o a semilinea ellip ic bounda y alue p oblem o he o m
Lw=γw − (x)w2in Ω ,
w= 0 on ∂Ω,(3.1)
whe e Lis a second o de uni o mly ellip ic ope a o o he o m (2.1) wi h coe icien s
sa is ying (2.2), γ∈R, and ∈C(Ω) sa is ies (x)>0 o each x∈Ω. I p > N and
w∈W2,p(Ω) ∩W1,p
0(Ω) is a posi i e solu ion o (3.1), hen
(L+ w)w=γw
and hanks o Theo em 2.1 we ha e ha
γ=σΩ
1[L+ w] (3.2)
and ha wis s ongly posi i e. The e o e, w(x)>0 o each x∈Ω and ∂nw(x)<0 o
each x∈∂Ω. The ollowing esul cha ac e izes he exis ence o posi i e solu ions o
(3.1).
Theo em 3.1. I p>N, hen he p oblem (3.1) possesses a posi i e solu ion in
W2,p(Ω) ∩W1,p
0(Ω) i , and only i , γ > σΩ
1[L]. Mo eo e , i is unique i i exis s.
Le θ[L,γ, ]deno e i . Then,
lim
γ↓σΩ
1[L]θ[L,γ, ]= 0 (3.3)
uni o mly in Ω.
Condi ion (3.3) says ha he posi i e solu ions bi u ca e om he i ial s a e w= 0
a he c i ical alue o he pa ame e γ=σΩ
1[L]. This esul is well known unde some
addi ional egula i y condi ions on he se e al coe icien s in ol ed in he model se ing,
e.g. see [14]. By he sake o comple eness we shall gi e a sho sel -con ained p oo o
i .
P oo o Theo em 3.1. Le wbe a posi i e solu ion o (3.1). Then, hanks o Theo em
2.1, we ha e (3.2) and hence Theo em 2.3(i) implies
γ=σΩ
1[L+ w]> σΩ
1[L].
The e o e, γ > σΩ
1[L] is necessa y o he exis ence o a posi i e solu ion. Assume
γ > σΩ
1[L]. I is easily seen ha la ge posi i e cons an s p o ide us wi h supe solu ions
o (3.1) and ha i ϕ > 0 s ands o he p incipal eigen unc ion associa ed wi h σΩ
1[L],
hen εϕ p o ide us wi h a bi a ily small posi i e subsolu ions i ε > 0 is su icien ly
small. The e o e, (3.1) possesses a leas a posi i e solu ion o each γ > σΩ
1[L]. We
poin ou ha he me hod o sub and supe solu ions wo ks ou hanks o he alidi y
o he s ong maximum p inciple.
16 M. DELGADO, J. L´
OPEZ-G´
OMEZ AND A. SU´
AREZ
is an eigen alue o a posi i e eigen unc ion, say ψ, o he second equa ion o (4.5). Since
τ1<0, (4.7) implies
σΩ
1[L1+ 2aθ[L1,λ,a]−λ−τ1]>0,
and he e o e, hanks o he s ong maximum p inciple, he i s equa ion o (4.5) wi h
τ=τ1possesses a unique solu ion. Namely,
u= (L1+ 2aθ[L1,λ,a]−λ−τ1)−1(bθ[L1,λ,a]ψ).
The e o e, unde condi ion (4.2) τ1<0 is an eigen alue o (4.5) and hence he s a e
(θ[L1,λ,a],0) is linea ly uns able. Finally i we assume (4.3), i is easily seen ha τ1= 0
is an eigen alue o (4.5) and ha any o he eigen alue has posi i e eal pa . The e o e,
unde condi ion (4.3) he s a e (θ[L1,λ,a],0) is linea ly neu ally s able.
The esul s conce ning wi h he o he semi- i ial s a e ollow by symme y in e ex-
changing L1,λ,aand bby L2,µ,dand c, espec i ely. ¤
By P oposi ion 4.1 we shall e e o he cu e (4.3) in he (λ, µ)-plane as he cu e o
change o s abili y o he semi- i ial posi i e solu ion (θ[L1,λ,a],0). Simila ly, he cu e
(4.4) will be e e eed as he cu e o change o s abili y o (0, θ[L2,µ,d]). The ollowing
esul p o ides us wi h he global beha io o hese cu es.
P oposi ion 4.2. The mapping F(λ)de ined by
F(λ) := σΩ
1[L2−c(x)θ[L1,λ,a]], λ > σΩ
1[L1],(4.8)
is con inuous s ic ly dec easing and sa is ies
lim
λ↓σΩ
1[L1]F(λ) = σΩ
1[L2],lim
λ↑∞ F(λ) = −∞.(4.9)
Simila ly, he mapping G(µ)de ined by
G(µ) := σΩ
1[L1−b(x)θ[L2,µ,d]], µ > σΩ
1[L2],(4.10)
is con inuous s ic ly dec easing and sa is ies
lim
µ↓σΩ
1[L2]G(µ) = σΩ
1[L1],lim
µ↑∞ G(µ) = −∞.(4.11)
P oo . The con inui y and mono onici y o F(λ) can be easily ob ained om Theo em
3.1, Co olla y 3.3 and Theo em 2.3(ii). The i s ela ion o (4.9) ollows om (3.3) and
Theo em 2.3(ii). We now show he second ela ion o (4.9). Since c∈C(Ω), c≥0,
c6= 0, he e exis s a ball Bwi h B⊂Ω such ha
cL:= min
B
c > 0.

SYMBIOTIC SPECIES 17
On he o he hand, by Theo em 3.4
lim
λ↑∞
θ[L1,λ,a]
λ=a−1uni o mly in B ,
and hence, he e exis s λ0such ha o λ > λ0
θ[L1,λ,a]>λ
2 maxBain B .
The e o e, Theo em 2.3 implies
F(λ)< σB
1[L2−c(x)θ[L1,λ,a]]< σB
1[L2]−cL
2 maxBaλ
o each λ > λ0. This comple es he p oo . The same a gumen shows he co esponding
p ope ies o G(µ). ¤
By P oposi ion 4.2 he cu es o change o s abili y o he semi- i ial posi i e solu-
ions mee a (σΩ
1[L1], σΩ
1[L2]). The nex esul p o ides us wi h he angen s o hese
cu es and hei conca i y o con exi y cha ac e a his co-dimension wo singula i y.
Lemma 4.3. Le ϕj, ϕ∗
jbe he p incipal eigen unc ions associa ed wi h Ljand L∗
j,
espec i ely, j= 1,2, whe e ∗s ands o he adjoin and
ZΩ
ϕ2
j= 1 ,ZΩ
ϕjϕ∗
j= 1 , j = 1 ,2.
Then,
θ[L1,λ,a]= (λ−σΩ
1[L1])m−1
a,1ϕ1+ (λ−σΩ
1[L1])2m−2
a,1U1+O((λ−σΩ
1[L1])3),
θ[L2,µ,d]= (µ−σΩ
1[L2])m−1
d,1ϕ2+ (µ−σΩ
1[L2])2m−2
d,1U2+O((µ−σΩ
1[L2])3),(4.12)
σΩ
1[L2−c(x)θ[L1,λ,a]] = σΩ
1[L2]−mc,a(λ−σΩ
1[L1]) −Mc,a(λ−σΩ
1[L1])2
+O((λ−σΩ
1[L1])3),
σΩ
1[L1−b(x)θ[L2,µ,d]] = σΩ
1[L1]−mb,d(µ−σΩ
1[L2]) −Mb,d(µ−σΩ
1[L2])2
+O((µ−σΩ
1[L2])3),
(4.13)
as λ↓σΩ
1[L1]and µ↓σΩ
1[L2], whe e
ma,1:= ZΩ
aϕ2
1ϕ∗
1>0, md,1:= ZΩ
dϕ2
2ϕ∗
2>0,
mc,a := m−1
a,1ZΩ
cϕ1ϕ2ϕ∗
2, mb,d := m−1
d,1ZΩ
bϕ2ϕ1ϕ∗
1,
18 M. DELGADO, J. L´
OPEZ-G´
OMEZ AND A. SU´
AREZ
Mc,a := ZΩ
c(x)(m−1
a,1ψ2ϕ1+m−2
a,1U1ϕ2)ϕ∗
2−mc,a ZΩ
ψ2ϕ∗
2.
Mb,d := ZΩ
b(x)(m−1
d,1ψ1ϕ2+m−2
d,1U2ϕ1)ϕ∗
1−mb,d ZΩ
ψ1ϕ∗
1,
and we ha e deno ed by βi,i= 1,2, and ψi,i= 1,2, he unique solu ions o he ollowing
linea p oblems in Ωunde homogeneous Di ichle bounda y condi ions
(L1−σΩ
1[L1])β1=ma,1ϕ1−a(x)ϕ2
1,ZΩ
β1ϕ1= 0 ,
(L2−σΩ
1[L2])β2=md,1ϕ2−d(x)ϕ2
2,ZΩ
β2ϕ2= 0 ,
(L1−σΩ
1[L1])ψ1= (−mb,d +m−1
d,1b(x)ϕ2)ϕ1,ZΩ
ψ1ϕ1= 0 ,
(L2−σΩ
1[L2])ψ2= (−mc,a +m−1
a,1c(x)ϕ1)ϕ2,ZΩ
ψ2ϕ2= 0 ,
U1:= β1−ma,2
ma,1·ϕ1, U2:= β2−md,2
md,1·ϕ2,
whe e
ma,2:= 2 ZΩ
aβ1ϕ1ϕ∗
1−ma,1ZΩ
β1ϕ∗
1, md,2:= 2 ZΩ
dβ2ϕ2ϕ∗
2−md,1ZΩ
β2ϕ∗
2.
P oo . The ela ions (4.12) ollow om he main heo em o [7] applied o (3.1) wi h
(L, γ, )=(L1, λ, a) and (L, γ, )=(L2, µ, d). Assume (L, γ, )=(L1, λ, a). Fo
λ≃σΩ
1[L1], he semi- i ial b anch (λ, θ[L1,λ,a]) may be pa ame ized by wo analy ic
unc ions
λ(s) = σΩ
1[L1] +
∞
X
j=1
λjsj, θ[L1,λ,a](s) = sϕ1+
∞
X
j=1
ujsj+1 , s ≃0,
whe e ZΩ
ujϕ1= 0 , j ≥1.(4.14)
Subs i u ing hese expansions in o (3.1) and iden i ying he e ms o o de wo and h ee
in syields
(L1−σΩ
1[L1])u1=λ1ϕ1−a(x)ϕ2
1in Ω , u1|∂Ω= 0 ,(4.15a)
(L1−σΩ
1[L1])u2=λ1u1+λ2ϕ1−2a(x)ϕ1u1in Ω , u2|∂Ω= 0 ,(4.15b)
espec i ely. F om (4.14) and he F edholm al e na i e applied o (4.15) i is easily seen
ha
λ1=ma,1, u1=β1, λ2=ma,2.
SYMBIOTIC SPECIES 19
To ob ain he i s ela ion o (4.12), i su ices calcula ing sas a unc ion o λ om
λ(s). Doing so, we ob ain ha
s(λ) = m−1
a,1(λ−σΩ
1[L1]) −ma,2
m3
a,1
(λ−σΩ
1[L1])2+O((λ−σΩ
1[L1])3).
Indeed, subs i u ing his expansion in o he expansion o θ[L1,λ,a](s), he i s ela ion
o (4.12) ge shown.
By s anda d pe u ba ion esul s (c . [18]), he p incipal eigen alues in he le hand
sides o (4.13) a y analy ically wi h λand µ. Thus, he e exis Kj∈R,j= 1 ,2, such
ha σΩ
1[L2−c(x)θ[L1,λ,a]] = σΩ
1[L2] + K1(λ−σΩ
1[L1])
+K2(λ−σΩ
1[L1])2+O((λ−σΩ
1[L1])3).(4.16)
Mo eo e , i Ψ(λ)>0 s ands o he p incipal eigen unc ion o σΩ
1[L2−c(x)θ[L1,λ,a]],
i.e. (L2Ψ(λ)−c(x)θ[L1,λ,a]Ψ(λ) =σΩ
1[L2−cθ[L1,λ,a]]Ψ(λ) in Ω
Ψ(λ) =0 on ∂Ω , (4.17)
no malized so ha
ZΩ
Ψ(λ)2= 1 ,ZΩ
(Ψ(λ)−ϕ2)ϕ2= 0 ,(4.18)
hen Ψ(λ) admi s a unique expansion o he o m
Ψ(λ) = Ψ0+ (λ−σΩ
1[L1])Ψ1+ (λ−σΩ
1[L1])2Ψ2+O((λ−σΩ
1[L1])3).(4.19)
Using (4.18) gi es
Ψ0=ϕ2,ZΩ
Ψjϕ2= 0 , j ≥1.(4.20)
Now, subs i u ing (4.16), (4.19) in o (4.17), using (4.12), (4.20) and iden i ying he
e ms wi h he same o de in λ−σΩ
1[L1], we ind ha
(L2−σΩ
1[L2])Ψ1= (K1+m−1
a,1c(x)ϕ1)ϕ2,(4.21)
(L2−σΩ
1[L2])Ψ2=c(x)(m−1
a,1ϕ1Ψ1+m−2
a,1U1ϕ2) + K1Ψ1+K2ϕ2.(4.22)
Applying F edholm’s al e na i e o (4.21) yields
K1=−m−1
a,1ZΩ
c(x)ϕ1ϕ2ϕ∗
2=−mc,a ,Ψ1=ψ2.
Now, subs i u ing hese alues in o (4.22) and applying F edholm’s al e na i e gi es
K2=−ZΩ
c(x)(m−1
a,1ϕ1ψ2+m−2
a,1U1ϕ2)ϕ∗
2+mc,a ZΩ
ψ2ϕ∗
2=−Mc,a .
20 M. DELGADO, J. L´
OPEZ-G´
OMEZ AND A. SU´
AREZ
By symme y, θ[L2,µ,d]and σΩ
1[L1−b(x)θ[L2,µ,d]] ha e he expansions gi en in he s a e-
men . The p oo is comple ed. ¤
By (4.13), he angen s o he cu es o change o s abili y o he semi- i ial posi i e
solu ions (4.3) and (4.4) a he singula i y (σΩ
1[L1], σΩ
1[L2]) a e gi en, espec i ely, by
he s igh lines
µ=σΩ
1[L2]−mc,a(λ−σΩ
1[L1]) , λ =σΩ
1[L1]−mb,d(µ−σΩ
1[L2]) .(4.23)
Close o he singula i y (σΩ
1[L1], σΩ
1[L2]) he con exi y o conca i y o hese cu es is
gi en by he sign o Mc,a and Md,b, espec i ely. Al hough in gene al he p oblem o
asce aining he sign o hese quan i ies migh be e y di icul o handle wi h, as hey
depend upon some unknown solu ions o ce ain homogeneous Di ichle bounda y alue
p oblems, he e a e some special cases whe e hese signs can be easily ound ou , as he
ollowing esul shows.
Lemma 4.4. I L1=L2is a sel adjoin ope a o and he coe icien s aand ca e
cons an s, hen
Mc,a >0.(4.24)
By symme y, i band da e cons an , hen
Mb,d >0.
The e o e, i a,b,cand da e cons an , hen he cu es o change o s abili y a e conca e
in a neighbo hood o (σΩ
1[L1], σΩ
1[L2]).
P oo . Since L1=L2is a sel adjoin ope a o , we ha e ha
ϕ1=ϕ2=ϕ∗
1=ϕ∗
2.
Hence,
ZΩ
ψ2ϕ∗
2=ZΩ
ψ2ϕ2= 0
and
Mc,a := cm−1
a,1ZΩ
ψ2ϕ2
1+cm−2
a,1ZΩ
U1ϕ2
1.(4.25)
Mo eo e ,
ma,1=aZΩ
ϕ3
1, ma,2= 2aZΩ
β1ϕ2
1, U1=β1−2RΩβ1ϕ2
1
RΩϕ3
1
ϕ1,(4.26)
and by he uniqueness o he solu ion o he co esponding bounda y alue p oblem in
he o hogonal complemen o ϕ1, we ind ha
ψ2=−c
a2RΩϕ3
1
β1.(4.27)
SYMBIOTIC SPECIES 21
Thus, subs i u ing (4.26) and (4.27) in o (4.25) gi es
Mc,a =−ca−2(ZΩ
ϕ3
1)−2(1 + c/a)ZΩ
β1ϕ2
1.(4.28)
To comple e he p oo o (4.24), i emains o show ha
ZΩ
β1ϕ2
1<0.(4.29)
Indeed, om he β1-equa ion i is easily seen ha
ZΩ
β1(L1−σΩ
1[L1])β1=−aZΩ
β1ϕ2
1,(4.30)
since RΩβ1ϕ1= 0. Mo eo e , β1changes o sign in Ω, and hence he a ia ional cha -
ac e iza ion o σΩ
1[L1] implies ha
ZΩ
β1(L1−σΩ
1[L1])β1>0.
The e o e, (4.30) implies (4.29). This comple es he p oo . ¤
In Figu e 1 we ha e ep esen ed he cu es o change o s abili y o he semi- i ial
posi i e solu ions in he case when a,b,cand da e cons an and L1=L2is sel adjoin .
λ
µ
µ=
λ=
σ
σΩ
Ω
1
1
F
G
[L
[L1
1
(λ)
(µ)
]
]
Figu e 1: The cu es o change o s abili y.

22 M. DELGADO, J. L´
OPEZ-G´
OMEZ AND A. SU´
AREZ
5. The exis ence o unbounded con inua o coexis ence s a es. Al hough wi h
less egula i y on he se e al coe icien s in ol ed in o ou se ing he abs ac heo y
o [21] applies o (1.1) i he solu ions o (1.1) a e ega ded as ixed poin s o a compac
ope a o on (C1
0(Ω))2. This obse a ion p o ides us wi h he ollowing esul , whe e he
no a ions in oduced in he p e ious sec ions will be kep .
Theo em 5.1. Fix λ > σΩ
1[L1]and ega d o µ∈Ras he bi u ca ion pa ame e .
Then, he poin
(µ, u, ) = (σΩ
1[L2−cθ[L1,λ,a]], θ[L1,λ,a],0)
is he only bi u ca ion poin o coexis ence s a es om he semi- i ial s a e (θ[L1,λ,a],0).
Mo eo e , he maximal componen (closed and connec ed) o coexis ence s a es emana -
ing om (θ[L1,λ,a],0) a µ=F(λ), say C+
(µ,u,0) ⊂R×C1
0(Ω) ×C1
0(Ω), is unbounded.
Now, ix µ < σΩ
1[L2]and ega d o λ∈Ras he bi u ca ion pa ame e . By P oposi ion
4.2 he e exis s a unique λµ> σΩ
1[L1]such ha µ=F(λµ). Then, he poin
(λ, u, ) = (λµ, θ[L1,λµ,a],0)
is he only bi u ca ion poin o coexis ence s a es om he cu e (θ[L1,λ,a],0). Mo eo e ,
he maximal componen (closed and connec ed) o coexis ence s a es emana ing om
(θ[L1,λ,a],0) a λ=λµ, say C+
(λ,u,0) ⊂R×C1
0(Ω) ×C1
0(Ω), is unbounded.
Simila ly, i we ix µ > σΩ
1[L2]and ega d o λ∈Ras he bi u ca ion pa ame e ,
hen he poin
(λ, u, ) = (σΩ
1[L1−bθ[L2,µ,d]],0, θ[L2,µ,d])
is he only bi u ca ion poin o coexis ence s a es om he semi- i ial s a e (0, θ[L2,µ,d])
and he maximal componen (closed and connec ed) o coexis ence s a es emana ing om
(0, θ[L2,µ,d])a λ=G(µ), say C+
(λ,0, )⊂R×C1
0(Ω) ×C1
0(Ω), is unbounded.
Finally, ix λ < σΩ
1[L1]and ega d o µ∈Ras he bi u ca ion pa ame e . By P opo-
si ion 4.2 he e exis s a unique µλ> σΩ
1[L2]such ha λ=G(µλ). In his case, he
poin
(µ, u, ) = (µλ,0, θ[L2,µλ,d])
is he only bi u ca ion poin o coexis ence s a es om he cu e (0, θ[L2,µ,d])and he max-
imal componen (closed and connec ed) o coexis ence s a es emana ing om (0, θ[L2,µ,d])
a µ=µλ, say C+
(µ,0, )⊂R×C1
0(Ω) ×C1
0(Ω), is unbounded.
P oo . The local bi u ca ions a e ob ained as an applica ion o he main heo em o
[7] using a he s anda d a gumen s. I emains o show ha each o he con inua o
coexis ence s a es emana ing om he semi- i ial s a es a e unbounded in he phase
space. We shall show his o he con inuum C+
(µ,u,0). The a gumen can be easily
adap ed o co e he emaining cases.
By Theo em 4.1 in [21] he con inuum C+
(µ,u,0) sa is ies some o he ollowing al e na-
i es: Ei he
SYMBIOTIC SPECIES 23
(i) C+
(µ,u,0) is unbounded in R×C1
0(Ω) ×C1
0(Ω); o
(ii) he e exis s µ∞∈Rsuch ha
λ=σΩ
1[L1−bθ[L2,µ∞,d]] (5.1)
and (µ∞,0, θ[L2,µ∞,d])∈closu e C+
(µ,u,0) ; o
(iii) he e exis s a posi i e solu ion ˆ
θ[L1,λ,a]6=θ[L1,λ,a]o
L1u=λu −au2in Ω , u|∂Ω= 0 ,(5.2)
such ha (σΩ
1[L1−bˆ
θ[L1,λ,a]], θ[L1,λ,a],0) ∈closu e C+
(µ,u,0) ; o
(i ) λ=σΩ
1[L1] and (σΩ
1[L2],0,0) ∈closu e C+
(µ,u,0) .
Since we a e assuming ha λ > σΩ
1[L1], al e na i e (i ) is no possible. Mo eo e ,
by Theo em 3.1 θ[L1,λ,a]is he unique posi i e solu ion o (5.2) and hence, al e na i e
(iii) is no possible ei he . No ice ha (5.1) is no possible ei he , since
σΩ
1[L1−bθ[L2,µ∞,d]]≤σΩ
1[L1].
The e o e, al e na i e (i) mus occu . This comple es he p oo . ¤
6. Coexis ence egions o small in e ac ion coe icien s. As an easy conse-
quence om Co olla y 3.3 we ob ain he ollowing esul .
Lemma 6.1. Assume ha
bMcM< aLdL,(6.1)
and ha (1.1) possesses a coexis ence s a e, say (u, ). Then,







λ >(c1)L
cMbM
aLdL
+σΩ
1[L1]µ1−cMbM
aLdL¶−bM
dL
(µ−(c2)L),
µ >(c2)L
cMbM
aLdL
+σΩ
1[L2]µ1−cMbM
aLdL¶−cM
aL
(λ−(c1)L),
(6.2)
and 






uM≤(λ−(c1)L)dL+ (µ−(c2)L)bM
aLdL−bMcM
,
M≤(µ−(c2)L)aL+ (λ−(c1)L)cM
aLdL−bMcM
.
(6.3)
P oo . F om (1.1) i is easily seen ha
u=θ[L1,λ+b ,a], =θ[L2,µ+cu,d].
24 M. DELGADO, J. L´
OPEZ-G´
OMEZ AND A. SU´
AREZ
Mo eo e , by Lemma 3.2 and Co olla y 3.3 we ha e
θ[L1,λ+b ,a]≤θ[L1,λ+bM M,aL]≤λ+bM M−(c1)L
aL
.
Thus,
uM≤λ+bM M−(c1)L
aL
.(6.4a)
Simila ly,
M≤µ+cMuM−(c2)L
dL
.(6.4b)
¿F om (6.4), ela ions (6.3) ollow eadily.
Mo eo e , he second ela ion o (6.3) implies
λ+bM M≤λaLdL+bMaL(µ−(c2)L)−cMbM(c1)L
aLdL−bMcM
,
and he e o e, since θ[L1,λ+bM M,aL]≥u > 0, we ind om Theo em 3.1 ha
λaLdL+bMaL(µ−(c2)L)−cMbM(c1)L
aLdL−bMcM
> σΩ
1[L1].(6.5a)
Simila ly,
µaLdL+cMdL(λ−(c1)L)−cMbM(c2)L
aLdL−bMcM
> σΩ
1[L2].(6.5b)
Rela ions (6.2) ollow eadily om (6.5). This comple es he p oo . ¤
No e ha i λand µsa is y (6.2), hen he ollowing ela ions hold





λ >(c1)L−bM
dL
(µ−(c2)L),
µ >(c2)L−cM
aL
(λ−(c1)L),
(6.6)
and he e o e, he igh hand sides o (6.3) a e posi i e. Indeed, i is easily seen om
Theo ems 2.2, 2.3 ha
σΩ
1[L1] = σΩ
1[L1−c1+c1]> σΩ
1[L1−c1]+(c1)L>(c1)L.(6.7)
Thus, we ind om (6.1) and (6.7) ha
cMbM
aLdL
(c1)L+σΩ
1[L1](1 −bMcM
aLdL
)>(c1)L,
SYMBIOTIC SPECIES 25
and hence,
(c1)L
cMbM
aLdL
+σΩ
1[L1]µ1−cMbM
aLdL¶−bM
dL
(µ−(c2)L)>(c1)L−bM
dL
(µ−(c2)L).
Simila ly,
(c2)L
cMbM
aLdL
+σΩ
1[L2]µ1−cMbM
aLdL¶−cM
aL
(λ−(c1)L)>(c2)L−cM
aL
(λ−(c1)L).
This shows he claim abo e.
Unde assump ion (6.1), (6.2) p o ides us wi h a simple eadily compu able necessa y
condi ion o he exis ence o a coexis ence s a e. Mo eo e , (6.3) shows ha we ha e a
p io i bounds in L∞(Ω) o he coexis ence s a es o (1.1) uni o mly on compac subse s
o he pa ame e space (λ, µ). By he Lp-es ima es o Agmon, Douglis and Ni enbe g we
ha e uni o m a p io i bounds in W2,p(Ω) o all p∈[2,∞). No ice ha he bounda y
o he non-exis ence egion gi en by (6.2) consis s o he s igh lines
λ= (c1)L
cMbM
aLdL
+σΩ
1[L1](1 −cMbM
aLdL
)−bM
dL
(µ−(c2)L),
µ= (c2)L
cMbM
aLdL
+σΩ
1[L2](1 −cMbM
aLdL
)−cM
aL
(λ−(c1)L).
In Figu e 2 we ha e ep esen ed hese lines oge he wi h he cu es o change o s abili y
o semi- i ial posi i e solu ions.
µ
λ
µ=
λ=
F
G
(λ)
(µ)
Figu e 2: Es ima ing he coexis ence egion.
32 M. DELGADO, J. L´
OPEZ-G´
OMEZ AND A. SU´
AREZ
and since his limi is posi i e and bounded away om ze o, he exis ence o α > 0
sa is ying (7.11) is easily ob ained om (7.23). This comple es he p oo . ¤
P oo o Theo em 7.1. (i) Assume (7.1), (7.2) and pick µ≥λ > σΩ
1[L]. I (1.1) possesses
a coexis ence s a e, say (u, ), hen we ind om he i s equa ion o (1.1) ha
λ=σΩ
1[L+au −b ]≤σΩ
1[L+aMu−bL ].(7.24)
Mo eo e , hanks o Lemma 7.2(i), we ind om (7.2) ha
u≤bM+dM
cL+aL
≤bL
aM
.
Thus,
aMu−bL ≤0,
and (7.24) gi es λ≤σΩ
1[L], which is impossible. The e o e, (1.1) can no admi a
coexis ence s a e. This comple es he p oo o Pa (i). Pa (ii) ollows by symme y,
in e exchanging he oles o λ,aand bby µ,dand c, espec i ely.
We now p o e (iii). Assume (7.1), (7.4) and ix λ < σΩ
1[L]. We a gue by con adic ion
assuming ha he e exis s a sequence o coexis ence s a es o (1.1), say (µn, un, n),
n≥1, such ha µn>max{µ0(λ),0},n≥1, and limn↑∞ µn=∞. Wi hou loss
o gene ali y we can assume ha µn≥λ o each n≥1. Le Ω1⊂Ω an a bi a y
subdomain o Ω wi h Ω1⊂Ω. By lemma 7.3(i), he e exis s α=α(Ω1)>0 such ha
o each n≥1 n
µn≥αin Ω1.
Mo eo e , by Lemma 7.2(i), we ha e ha o each n≥1
un
µn≤bM+dM
cL+aL
n
µn
.
Thus, by (7.4) he e exis s ε > 0 such ha o each n≥1
un
µn≤bL
aM
n
µn−εin Ω1.
Hence,
aMun−bL n≤ −εaMµnin Ω1∀n≥1.(7.25)
On he o he hand, we ind om he i s equa ion o (1.1) ha
λ=σΩ
1[L+aun−b n]≤σΩ1
1[L+aMun−bL n]
and he e o e, (7.25) gi es
λ≤σΩ1
1[L]−εaMµn↓ −∞ as n→ ∞.
This con adic ion shows ha (1.1) does no admi a coexis ence s a e o µla ge and
comple es he p oo o his pa . Pa (i ) ollows by symme y. ¤

SYMBIOTIC SPECIES 33
7.2. A p io i bounds o N≤5.The ollowing esul p o ides us wi h uni o m a
p io i bounds in L∞ o he coexis ence s a es o (1.1).
Theo em 7.4. Unde condi ion (7.1), i N≤5,bLcL> aMdMand o some α > 0
max {|λ|,|µ|} ≤ α ,
hen he e exis s a cons an C=C(α, Ω, a, b, c, d)such ha
kukL∞(Ω) ≤C , k kL∞(Ω) ≤C ,
o any coexis ence s a e (u, )o (1.1).
This esul is op imal in he sense ha i N > 5, hen he e a e choices o he
se e al coe icien s and o Ω o which he uni o m a p io i bounds a e los (c . he inal
commen s in Sec ion 5 o [21] and Theo em 1.4 o [27]). Fo ins ance, i a,b,c,da e
cons an s and λ=µ, hen o any coexis ence s a e (u, ) o (1.1) i is easily seen ha
(L−λ+au +d )((b+d) −(c+a)u) = 0
and hence,
=c+a
b+du , (7.26)
since σΩ
1[L−λ+au +d ]>0. The e o e, (u, ) is a coexis ence s a e o (1.1) i , and
only i , (7.26) holds and uis a posi i e solu ion o
Lu=λu +bc −ad
b+du2in Ω , u|∂Ω= 0 .(7.27)
I bc < ad, hen he coe icien o u2in (7.27) is nega i e and hence he posi i e solu ions
o (7.27) possesses uni o m a p io i bounds on compac subin e als o λ. On he
con a y, when bc > ad he coe icien o u2in (7.27) is posi i e and he e o e (7.27) is
a supe linea p oblem. In his case i is well known ha a p io i bounds a e a ailable i
2<N+2
N−2(c . [13]), i.e. i N≤5, while in he case when N≥6 he a p io i bounds a e
in gene al los and he s uc u e o he se o posi i e solu ions can change d as ically
as ei he he geome y o Ω changes o he spa ial dimension Ninc eases. Being he
highe dimensional case ou side he scope o his wo k we send o he in e es ed eade
in u he de ails o [4] and [8].
In he special case when L=−∆ and a,b,cand da e cons an s Theo em 7.4 is
gi en by Lemma 4.3 o [27], bu he p oo o [27] can no be adap ed o co e ou
cu en si ua ion he e, as i will become clea la e . The main di icul y coming om
he ac ha now he coe icien s a e no cons an . To p o e Theo em 7.4 we will a gue
by con adic ion using he blowing up a gumen in oduced in [13] o he case o one
single equa ion. I should be no ed ha ou blowing up a gumen is somewha di e en
om he co esponding a gumen used in [27].
34 M. DELGADO, J. L´
OPEZ-G´
OMEZ AND A. SU´
AREZ
P oo o Theo em 7.4. We shall p o e he esul in case λ≥µ. By symme y, he esul
is also ue when µ≥λ. I he conclusion o Theo em 7.4 is alse, hen he e exis s a
sequence o coexis ence s a es (λk, µk, uk, k), k≥1, wi h −α≤µk≤λk≤α, such
ha
lim sup
k→∞
(kukkL∞(Ω) +k kkL∞(Ω)) = ∞.(7.28)
We claim ha
lim sup
k→∞ kukkL∞(Ω) = lim sup
k→∞ k kkL∞(Ω) =∞.(7.29)
Indeed, i {k kkL∞(Ω)}k≥1is bounded by some posi i e cons an β, hen we ind om
he i s equa ion o (1.1) ha
Luk≤(α+bMβ)uk−au2
k
and he e o e, i ollows om Lemma 3.2 and Co olla y 3.3, ha {kukkL∞(Ω)}k≥1is
also bounded. By (7.28) his is impossible. Simila ly, i {kukkL∞(Ω)}k≥1is bounded,
hen {k kkL∞(Ω)}k≥1is also bounded. The e o e, (7.29) is sa is ied. By chosing a
subsequence, i necessa y, we can assume ha
lim
k→∞ kukkL∞(Ω) =∞,lim
k→∞(λk, µk) = (λ∞, µ∞),(7.30)
o some (λ∞, µ∞)∈R2sa is ying −α≤µ∞≤λ∞≤α. No e ha hanks o Lemma
7.2(ii) we ha e ha
k≤cM+aM
bL+dL
uk∀k≥1.(7.31)
Fo each k≥1, pick xk∈Ω such ha
Mk:= uk(xk) = kukkL∞(Ω) .(7.32)
Since Ω is bounded, wi hou loss o gene ali y we can assume ha
lim
k→∞ xk=x∞∈Ω.(7.33)
Now, we conside wo di e en si ua ions, acco dingly wi h whe he x∞∈Ω o x∞∈
∂Ω.
Assume ha x∞∈Ω. Then,
δ:= d(x∞, ∂Ω)/2>0.
Mo eo e , se ing
ρk:= M−1/2
k, k ≥1,
SYMBIOTIC SPECIES 35
we ha e limk→∞ ρk= 0, since hanks o (7.30) and (7.32) limk→∞ Mk=∞. Now, i is
easily seen ha he change o a iables
y:= x−xk
ρk
,(zk, wk) := ρ2
k(uk, k), k ≥1,(7.34)
ans o ms he sys em o (1.1) in o
Akzk=ρ2
kλkzk−a(xk+ρky)z2
k+b(xk+ρky)zkwk,
Akwk=ρ2
kµkwk−d(xk+ρky)w2
k+c(xk+ρky)zkwk,(7.35)
whe e
Ak=−
N
X
i,j=1
aij(xk+ρky)∂i∂j+ρk
N
X
j=1
bj(xk+ρky)∂j+ρ2
ke(xk+ρky),(7.36)
p o ided xk+ρky∈Ω. By de ini ion o δ, o ksu icien ly la ge, |x−xk| ≤ δimplies
x=xk+ρky∈Ω. Hence, |y| ≤ δ
ρkimplies x=xk+ρky∈Ω and so (7.35) holds.
Since limk→∞ δ
ρk=∞, gi en R > 0 a bi a y BR⊂Bδ/ρk o ksu icien ly la ge, whe e
o any τ > 0Bτs ands o he ball o adius τcen e ed a he o igin. Now, om he
de ini ion o ρkwe ha e ha
zk=ρ2
kuk=uk
Mk
and hence,
kzkkL∞(BR)= 1 , zk(0) = 1 ,∀k≥1.(7.37)
Mo eo e , hanks o (7.31) and (7.37), we ind ha
kwkkL∞(BR)≤cM+aM
bL+dL∀k≥1.(7.38)
Now he same compac ness a gumen o he p oo o Theo em 1.1 in [13] shows ha
gi en any p > N and passing o a sui able subsequence, again elabeled by k, he e
exis s (z, w)≥(0,0) in W2,p(BR)∩C1,ν (BR), 0 < ν < 1, such ha
lim
k→∞(zk, wk) = (z, w) in (W2,p(BR)∩C1,ν(BR))2.
By H¨olde con inui y z(0) = 1. Mo eo e , passing o he limi as k→ ∞ in (7.35) gi es
−
N
X
i,j=1
aij(x∞)∂i∂jz=−a(x∞)z2+b(x∞)zw ,
−
N
X
i,j=1
aij(x∞)∂i∂jw=−d(x∞)w2+c(x∞)zw ,
(7.39)
36 M. DELGADO, J. L´
OPEZ-G´
OMEZ AND A. SU´
AREZ
in BR, o any R > 0. By a s anda d diagonal sequence a gumen i is easily seen ha
z,w∈W2,p
loc (RN) and ha (7.39) holds ue in he whole o RN. Mo eo e , s anda d
ellip ic egula i y heo y implies ha z,w∈C2(RN). Fu he mo e, by a linea change
o coo dina es (c . [13] pg. 890), (7.39) can be educed o
−∆z=−a(x∞)z2+b(x∞)zw
−∆w=−d(x∞)w2+c(x∞)zw in RN.(7.40)
¿F om (7.40), i is easily seen ha
(−∆ + a(x∞)z+d(x∞)w)(w−c(x∞) + a(x∞)
b(x∞) + d(x∞)z) = 0 .
Since (z, w)≥(0,0) and z(0) = 1, he po en ial
V:= a(x∞)z+d(x∞)w
sa is ies V≥0 and V6= 0. The e o e, due o he ollowing lemma, whose p oo we
pos pone up o conclude he p oo o Theo em 7.4, we ind ha
w=c(x∞) + a(x∞)
b(x∞) + d(x∞)z . (7.41)
Lemma 7.5. Assume ha ei he D=RNo D=RN
+, whe e
RN
+={x∈RN:xN≥0}.
I V∈L∞(D)∩Cν(D),V≥0,V6= 0, hen θ= 0 is he only bounded solu ion o
(−∆ + V)θ= 0 in D . (7.42)
Subs i u ing (7.41) in o he i s equa ion o (7.40) and ea anging e ms gi es
−∆z=b(x∞)c(x∞)−a(x∞)d(x∞)
b(x∞) + d(x∞)z2in RN.(7.43)
Since bLcL> aMdM,b(x∞)c(x∞)> a(x∞)d(x∞) and hence, hanks o Theo em 1.1
o [13], z= 0 is he unique non-nega i e solu ion o (7.43), because N≤5. This is a
con adic ion wi h z(0) = 1. The e o e, x∞∈∂Ω. Now, he same a gumen as in Case
2 o he p oo o Theo em 1.1 in [13] shows ha he p oblem
−∆z=−a(x∞)z2+b(x∞)zw
−∆w=−d(x∞)w2+c(x∞)zw in RN
+.(7.44)
SYMBIOTIC SPECIES 37
possesses a non-nega i e solu ion couple (z, w) wi h z(0) = 1. The same a gumen as
abo e shows ha his is impossible. This con adic ion shows he exis ence o uni o m
a p io i bounds and comple es he p oo o he heo em. ¤
We now p o e Lemma 7.5, which is a Liou ille ype esul in e es ing in i s own igh .
In he p oo we use he concep s and esul s in Chap e 4 o [31].
P oo o Lemma 7.5. Thanks o Theo em 3.3(iii) in page 148 o [31], he Sch ¨odinge
ope a o ∆ −Vis subc i ical on D, i.e. i possesses a G een unc ion G(x, y) on D.
The e o e, hanks o Theo em 3.8(i) in page 151 o [31] o each non-nega i e p∈Cν
0(D),
p6= 0, he e exis s posi i e solu ions u∈C2,ν(D) o
(−∆ + V)u=p . (7.45)
Mo eo e , (7.45) possesses a minimal solu ion u0, gi en by
u0(x) = ZD
G(x, y)p(y)dy ,
and any o he solu ion o (7.45) mus be gi en by
u=u0+θ ,
o some some posi i e solu ion θo (7.42). The minimali y o u0shows ha θ= 0 is
he unique solu ion o (7.42). This comple es he p oo . ¤
Rema k 7.6. (a) Al hough (7.31) implies w≤cM+aM
bL+dLz, his does no necessa ily en ails
w≤c(x∞) + a(x∞)
b(x∞) + d(x∞)z(7.46)
and hence, Lemma 4.5 o [27] can no be applied o show ha (z, w) = (0,0) is he
unique solu ion o (7.40). In ac , ou co esponding Liou ille ype esul is subs an ially
sha pe han Lemma 4.5 o [27], as we do no need assuming (7.46) o in e z=w= 0.
(b) By he Lpes ima es o Agmon, Douglis & Ni enbe g and Mo ey’s Theo em,
Theo em 7.4 p o ides us wi h a uni o m a p io i bounds in C1
0(Ω) ×C1
0(Ω) o he
coexis ence s a es o (1.1) on any compac subse o he (λ, µ)-plane.
7.3. On he exis ence o coexis ence s a es in case N≤5.As an immedia e
consequence, om Theo em 5.1, Theo em 7.1 and Theo em 7.4 we ob ain he ollowing
esul .
Theo em 7.7. (i) I N≤5,(7.4) and
λ < σΩ
1[L−bθ[L,µ,d]],(7.47)
a e sa is ied, hen (1.1) possesses a coexis ence s a e.

38 M. DELGADO, J. L´
OPEZ-G´
OMEZ AND A. SU´
AREZ
(ii) I N≤5,(7.5) and
µ < σΩ
1[L−cθ[L,λ,a]],(7.48)
a e sa is ied, hen (1.1) possesses a coexis ence s a e.
(iii) I N≤5and ei he (7.4) o (7.5) is sa is ied, hen (1.1) possesses a coexis ence
s a e p o ided
λ < σΩ
1[L], µ < σΩ
1[L].(7.49)
P oo . We i s show Pa (i). Fix λ < σΩ
1[L] and conside µas he main bi u ca ion
pa ame e . By Theo em 7.1 (iii) he e exis s µ=µ(λ) such ha λ > σΩ
1[L−bθ[L,µ(λ),d]]
and (1.1) does no admi a coexis ence s a e o µ > µ(λ).
Mo eo e , by Theo em 5.1 he con inuum C+
(µ,0, )o coexis ence s a es emana ing
om (0, θ[L,µ,d]) a µλis unbounded, whe e µλis he unique alue o µ>σΩ
1[L] o
which λ=σΩ
1[L− bθ[L,µ,d]]. Fu he mo e, (7.4) implies bLcL> aMdMand hence, we
conclude om Theo em 7.4 ha (1.1) possesses a coexis ence s a e o each µ < µλ.
This comple es he p oo o Pa (i). Pa (ii) ollows by symme y and Pa (iii) is an
easy consequence om Pa s (i), (ii). ¤
In p ac ice, he e i ica ion o condi ions (7.47) and (7.48) is a om easy, as each
o hem in ol es he e alua ion o he p incipal eigen alue o a second o de ellip ic op-
e a o whose associa ed po en ial is gi en h ough by a posi i e solu ion o a semilinea
ellip ic bounda y alue p oblem. The nex esul s p o ide us wi h some easily com-
pu able su icien condi ions in e ms o he se e al coe icien s in ol ed in he se ing o
(1.1) so ha (7.47), o (7.48), holds. Ou analysis ex ends o he case o gene al second
o de ellip ic ope a o s he es ima es o Theo em 2.3 (c) in [26], ound o he special
case o ope a o s in di e gence o m.
Lemma 7.8. Assume ha Lis a di e en ial ope a o o he o m (2.1) whose coe -
icien s sa is y (2.2). Fo γ > σΩ
1[L], le θ[L,γ, ]deno e he posi i e solu ion o (3.1).
Then, he e exis s a posi i e cons an
K=K(L, , Ω) ≥max ½kϕk∞
m ,1
,1
L¾(7.50)
such ha
kθ[L,γ, ]k∞≤K(γ−σΩ
1[L]) ∀γ≥σΩ
1[L],
whe e ϕis he p incipal eigen unc ion associa ed wi h L, no malized so ha
ZΩ
ϕ2= 1 ,
and m ,1is he cons an de ined in he s a emen o Lemma 4.3.
P oo . Thanks o Lemma 4.3
dθ[L,γ, ]
dγ c{γ=σΩ
1[L]}=ϕ
m ,1
,
SYMBIOTIC SPECIES 39
and hence, he e exis δ > 0 and a cons an C > 0 such ha
kθ[L,γ, ]k∞≤C(γ−σΩ
1[L]) (7.51)
o each γ∈[σΩ
1[L], σΩ
1[L] + δ].
On he o he hand, i ollows om Co olla y 3.3 ha
θ[L,γ, ]≤γ−eL
L
.
Thus, he e exis s a cons an ˆ
C > 0 such ha
kθ[L,γ, ]k∞
γ−σΩ
1[L]≤γ−eL
γ−σΩ
1[L]·1
L≤ˆ
C1
L
o each γ≥σΩ
1[L] + δ. This comple es he p oo . ¤
Theo em 7.9. Assume ha Lis a di e en ial ope a o o he o m (2.1) whose coe i-
cien s sa is y (2.2), and le K1:= K(L, a, Ω),K2:= K(L, d, Ω) deno e he wo cons an s
whose exis ence was shown by Lemma 7.8. Then, he ollowing asse ions a e ue:
(i) I N≤5,(7.4) and
λ < σΩ
1[L], λ < min{σΩ
1[L]−bMK2(µ−σΩ
1[L]) , σΩ
1[L]−bM
dL
(µ−eL)}
a e sa is ied, hen (1.1) possesses a coexis ence s a e.
(ii) I N≤5,(7.5) and
µ < σΩ
1[L], µ < min{σΩ
1[L]−cMK1(λ−σΩ
1[L]) , σΩ
1[L]−cM
aL
(λ−eL)}
a e sa is ied, hen (1.1) possesses a coexis ence s a e.
P oo . By Lemma 7.8, we ha e ha
kθ[L,λ,a]k∞≤K1(λ−σΩ
1[L]) ,kθ[L,µ,d]k∞≤K2(µ−σΩ
1[L]) .
Thus, i ollows om Theo em 2.3 ha
σΩ
1[L−bθ[L,µ,d]]≥σΩ
1[L−bMkθ[L,µ,d]k∞]≥σΩ
1[L−bMK2(µ−σΩ
1[L])]
=σΩ
1[L]−bMK2(µ−σΩ
1[L]) .
Simila ly,
σΩ
1[L−cθ[L,λ,a]]≥σΩ
1[L]−cMK1(λ−σΩ
1[L]) .
On he o he hand, Co olla y 3.3 implies
θ[L,λ,a]≤λ−eL
aL
, θ[L,µ,d]≤µ−eL
dL
,
and he same a gumen as abo e shows ha
σΩ
1[L−cθ[L,λ,a]]≥σΩ
1[L]−cM
aL
(λ−eL),
σΩ
1[L−bθ[L,µ,d]]≥σΩ
1[L]−bM
dL
(µ−eL).
Theo em 7.7 comple es he p oo . ¤
40 M. DELGADO, J. L´
OPEZ-G´
OMEZ AND A. SU´
AREZ
8. The maximum p inciple. Mul iplici y esul s. In his sec ion we use he
abs ac heo y o [2] o show ha he me hod o sub and supe solu ions is alid o
(1.1). Then, we use i o analyze he s uc u e o he se o λ’s (o µ’s) o which (1.1)
possesses a coexis ence s a e and o ge some mul iplici y esul s o coexis ence s a es.
The basic echnical ool o p o e hese esul s is he s ong maximum p inciple o linea
coope a i e sys ems. The alidi y o he s ong maximum p inciple is gua an eed i , o
ins ance, we assume ha
b(x)>0, c(x)>0,∀x∈Ω.(8.1)
So, o he es o his sec ion we shall assume ha his condi ion is sa is ied.
8.1. The s ong maximum p inciple o coope a i e sys ems. I (u0, 0) is a
coexis ence s a e o (1.1), hen i s linea ized s abili y is gi en by he eigen alues o he
linea iza ion o (1.1) a (u0, 0), i.e. by he τ’s o which he ollowing p oblem has some
solu ion (u, )∈W2,p
0(Ω) ×W2,p
0(Ω), (u, )6= (0,0), p > N,
µL10
0L2¶µu
¶=Aµu
¶+τµu
¶,(8.2)
whe e
A=µλ−2au0+b 0bu0
c 0µ−2d 0+cu0¶.(8.3)
No e ha hanks o (8.1) he o -diagonal en ies o his ma ix a e posi i e and so
he coupling ma ix Ais o coope a i e ype. Mo e gene ally, we conside he linea
coope a i e eigen alue p oblem (8.2) wi h (u, )∈W2,p
0(Ω) ×W2,p
0(Ω) o some p>N
and
A=µα(x)β(x)
γ(x)ρ(x)¶,(8.4)
whe e α,β,γ,ρ∈C(Ω) and he o -diagonal en ies, βand γ, a e posi i e almos
e e ywhe e in Ω. In he sequel we se
L:= µL10
0L2¶−A(8.5)
and suppose ha p>N. Now, o s a e he maximum p inciple we need some o
no a ion. Gi en (u, )∈Lp(Ω) ×Lp(Ω), i is said ha (u, )≥0 i u≥0 and ≥0.
I in addi ion u6= 0 o 6= 0, hen i is said ha (u, )>0. A couple (u, )∈
W2,p
0(Ω) ×W2,p
0(Ω) is said o be s ongly posi i e i u(x)>0, (x)>0 o all x∈Ω
and ∂nu(x)<0, ∂n (x)<0 o all x∈∂Ω, whe e nis he ou wa d uni no mal a x.
De ini ion 8.1. The ope a o Lde ined by (8.5) is said o sa is y he s ong maximum
p inciple in Ωi x:= (u, )∈W2,p
0(Ω) ×W2,p
0(Ω) and Lx > 0imply ha xis s ongly
posi i e.
SYMBIOTIC SPECIES 41
De ini ion 8.2. A unc ion x:= (u, )∈W2,p(Ω) ×W2,p(Ω) is said o be a supe so-
lu ion o Lin Ωi x|∂Ω≥0and Lx≥0. I in addi ion Lx > 0, o x|∂Ω>0, hen i is
said ha xis a s ic supe solu ion.
Now, using Theo ems 2.1, 2.2 o Sec ion 2, he p oo o Theo em 2.1 in [24] can
be easily adap ed o co e ou gene al se ing p o iding us wi h he ollowing gene al
e sions o Theo ems 2.1, 2.2 o Sec ion 2.
Theo em 8.3. The e exis s a leas eigen alue o (8.2), deno ed by σΩ
1[L]and called
p incipal eigen alue o Lin Ω. This eigen alue is simple and possesses a unique eigen-
unc ion, up o mul iplica i e cons an s, which can be aken posi i e, he so called p in-
cipal eigen unc ion o Lin Ω. Mo eo e , he p incipal eigen unc ion is s ongly posi i e
and σΩ
1[L]is he only eigen alue o (8.2) possessing a posi i e eigen unc ion. Fu he -
mo e, any o he eigen alue σo (8.2) sa is ies
Re σ > σΩ
1[L]
and (L+ν)−1∈ L(Lp(Ω)×Lp(Ω)) is posi i e, compac and i educible o ν > −σΩ
1[L].
Theo em 8.4. The ollowing asse ions a e equi alen :
(i) σΩ
1[L]>0;
(ii) Lpossesses a posi i e s ic supe solu ion in W2,p(Ω) ×W2,p(Ω);
(iii) Lsa is ies he s ong maximum p inciple.
Mo eo e , he ollowing gene alized maximum p inciple holds.
Theo em 8.5. I Lsa is ies he s ong maximum p inciple, hen any s ic supe so-
lu ion x:= (u, )∈W2,p(Ω) ×W2,p(Ω) o Lis posi i e in Ω. In ac , u(x)>0and
(x)>0 o all x∈Ω. I will simply said ha Lsa is ies he gene alized maximum
p inciple in Ω.
P oo . I is based upon Theo em Ano [34]. Thanks o Theo em 8.4, σΩ
1[L]>0. Le
h > 0 deno e he p incipal eigen unc ion associa ed wi h σΩ
1[L]>0. We ha e ha
Lh > 0 in Ω. The e o e, hanks o Theo em Ano [34], some o he ollowing op ions
occu s: Ei he (i) x > 0 in Ω, o (ii) x= 0 in Ω, o (iii) x=αh o some α < 0. Since,
we a e assuming ha xis a s ic supe solu ion, he op ions (ii) and (iii) a e excluded.
The e o e, x > 0 in Ω. Co olla y 2 o [34] comple es he p oo . ¤
Thanks o hese esul s, o any ope a o Lo he ype (8.5) he e exis s ωsuch ha
L+νsa is ies he he gene alized maximum p inciple o all ν > ω. The e o e, he p oo
o Theo em 9.4 o [2] ca ies o e mu a is mu andis o ou p esen si ua ion, showing
ha he me hod o sub and supe solu ions wo ks ou o he nonlinea model (1.1). To
s a e ou esul we need o in oduce he concep o sub and supe solu ion.
De ini ion 8.6. A posi i e unc ion x= (u, )∈W2,p(Ω) ×W2,p(Ω) is said o be a
subsolu ion o (1.1) i
L1u≤λu −a(x)u2+b(x)u
L2 ≤µ −d(x) 2+c(x)u in Ω,
48 M. DELGADO, J. L´
OPEZ-G´
OMEZ AND A. SU´
AREZ
Lemma 8.13. (i) Le (µ, u, ) = (µ0, u0, 0)be a coexis ence s a e o (1.1) such ha
σΩ
1[Lµ0]>0,(8.16)
whe e Lµ0is he ope a o de ined by (8.5) wi h A(x)gi en by (8.3). Then, he e ex-
is s ε > 0and a di e en iable mapping (u, ):(µ0−ε, µ0+ε)→P2such ha
(u(µ0), (µ0)) = (u0, 0)and (µ, u(µ), (µ)) is a coexis ence s a e o (1.1) o each
µ∈(µ0−ε, µ0+ε). Mo eo e , he mapping µ→(u(µ), (µ)) is s ic ly inc easing and
he e exis s a neighbo hood Qo (µ0, u0, 0)in R×(Ce(Ω))2such ha i (µ, u, )∈ Q
is a solu ion o (1.1), hen (u, ) = (u(µ), (µ)).
(ii) Assume σΩ
1[Lµ0] = 0, ins ead o (8.15), and le Φdeno e he p incipal eigen-
unc ion associa ed wi h σΩ
1[Lµ0]. Then, he e exis s ε > 0and a di e en iable mapping
(µ, u, ) : (−ε, ε)→R×P2such ha (µ(0), u(0), (0)) = (µ0, u0, 0)and o each
s∈(−ε, ε) (µ(s), u(s), (s)) is a coexis ence s a e o (1.1). Mo eo e ,
µ(s) = µ0+ ˆµ(s),(u(s), (s)) = (u0, 0) + sΦ + (ˆu(s),ˆ (s)) ,(8.17)
whe e ˆµ(s) = 0(s),ˆu(s) = o(s)and ˆ (s) = o(s)as s→0, and he e exis s a neighbo hood
Qo (µ0, u0, 0)in R×(Ce(Ω))2such ha i (µ, u, )∈ Q is a solu ion o (1.1), hen
(µ, u, ) = (µ(s), u(s), (s))
o some s∈(−ε, ε). Fu he mo e,
sgn µ0(s) = sgn σΩ
1[Ls],(8.18)
whe e
Ls=µL10
0L2¶−µλ−2au(s) + b (s)bu(s)
c (s)µ(s)−2d (s) + cu(s)¶.
I σΩ
1[Lµ]>0, hen he Le ay-Schaude o mula implies ha he local index
i(Kµ,(uµ, µ)) = 1
and he e o e, hanks o Lemma 8.11, (1.1) mus ha e a u he coexis ence s a e.
The e o e, in his case he p oo is comple ed.
Now, assume ha σΩ
1[Lµ] = 0 and le (µ(s), u(s), (s)) deno e he cu e o coexis ence
s a es h ough by (µ, uµ, µ), o s= 0, whose exis ence is gua an eed by Lemma 8.13.
Since Φ >0, (u(s), (s)) is s ic ly inc easing and hence, i µ(s) = µ o some s6= 0,
hen (1.1) possesses wo coexis ence s a es. Namely, (uµ, µ) and (u(s), (s)). Thus,
wi hou loss o gene ali y we can assume ha
µ(s)6=µ∀0<|s|< ε . (8.19)

SYMBIOTIC SPECIES 49
We claim ha
µ(s)< µ ∀s∈(−ε, 0) .(8.20)
Indeed, i he e exis s s1<0 such ha µ1:= µ(s1)≥µ, hen
(u(s1), (s1)) <(u(0), (0)) = (uµ, µ)≤(uµ1, µ1),(8.21)
since (u(s), (s)) is inc easing in sand he minimal solu ion is non-dec easing in µ. He e,
(uµ1, µ1) s ands o he minimal coexis ence s a e o (1.1) o µ=µ1. Rela ion (8.21)
con adic s he minimali y o (uµ1, µ1). Thus, (8.20) ge shown. Mo eo e , by (8.19),
ei he µ(s)< µ o all s∈(0, ε), o µ(s)> µ o all s∈(0, ε), so we can dis inguish wo
cases:
Case a: Assume ha µ(s)< µ o all s∈(0, ε). Then, since µ < µ∗and (1.1)
possesses a coexis ence s a e o each alue o he pa ame e in [µ, µ∗], he e exis s a
sequence o coexis ence s a es (µn, un, n), n≥1, such ha limn→∞ µn=µand µn> µ
o all n≥1. By he exis ence o uni o m a p io i bounds, wi hou loss o gene ali y we
can assume ha
lim
n→∞(un, n) = (u0, 0),
o some non-nega i e solu ion (u0, 0) o (1.1). Since λ < σΩ
1[L1] and µ > µλ, wi h a
simila a gumen as in he p oo o Theo em 8.8, i is easily seen ha (µ, u0, 0) is a
coexis ence s a e. Mo eo e , by he uniqueness ob ained as an applica ion o Lemma
8.13(ii), (µn, un, n)6∈ Q o each n≥1 and hence, (µ, u0, 0)6∈ Q. In pa icula ,
(µ, u0, 0)6= (µ, uµ, µ) and he e o e, (1.1) possesses a leas wo coexis ence s a es.
Case b: Now, assume ha
µ(s)> µ ∀s∈(0, ε).(8.22)
Then, hanks o Lemma 8.13(ii), (µ, uµ, µ) is an isola ed solu ion o (1.1) and so
i(Kµ,(uµ, µ)) is well de ined. By Lemma 8.11, o comple e he p oo o Theo em
8.10, i su ices o show ha
i(Kµ,(uµ, µ)) = 1 .(8.23)
By (8.22) he e exis s s1∈(0, ε) o which µ0(s1)>0. By (8.18), σΩ
1[Ls1]>0 and he e-
o e, we ind om Theo em 8.3 and he linea ized s abili y p inciple ha (u(s1), (s1))
is exponen ially asymp o ically s able. Thus, Le ay-Schaude ’s o mula implies
i(Kµ(s1),(u(s1), (s1))) = 1 .(8.24)
Since (µ(s1), u(s1), (s1)) is non-degene a e and s→(u(s), (s)) is inc easing he e
exis s δ > 0 such ha i
ρ1:= k(u(s1), (s1))ke−δ , ρ2:= k(uµ, µ)ke−δ ,
hen (1.1) does no admi a coexis ence s a e in
[µ(s1), µ(s1) + δ]×∂(Pρ1 Pρ2).
50 M. DELGADO, J. L´
OPEZ-G´
OMEZ AND A. SU´
AREZ
Mo eo e , by he uniqueness o Lemma 8.13(ii), δ > 0 can be chosen so ha (1.1) does
no ha e a coexis ence s a e in Pρ1 Pρ2 o µ=µ(s1) + δei he . Thus, he homo opy
in a iance implies
i(Kµ(s1), Pρ1 Pρ2) = 0 .(8.25)
Now, o δ > 0 su icien ly small se
ρ:= k(u(s1), (s1)ke+δ .
By (8.24), (8.25), we ind ha
i(Kµ(s1), Pρ Pρ2) = 1 .
Mo eo e , by he mono onici y o (u(s), (s)) and he uniqueness gi en by Lemma
8.13(ii), (1.1) does no admi a coexis ence s a e on
[µ, µ(s1)] ×∂(Pρ Pρ2).
This implies (8.23) and comple es he p oo o he heo em. ¤
Simila ly, o he case o small in e ac ion coe icien s we ha e he ollowing esul .
Theo em 8.14. Assume (6.1). Then ollowing asse ions a e ue:
(i) Assume µ>σΩ
1[L2]and Λ=[λ∗,∞)wi h λ∗< σΩ
1[L1−b(x)θ[L2,µ,d]]. Then,
(1.1) possesses a leas wo coexis ence s a es o each λ∈(λ∗, σΩ
1[L1−b(x)θ[L2,µ,d]]).
(ii) Assume λ > σΩ
1[L1]and M= [µ∗∞)wi h µ∗< σΩ
1[L2−c(x)θ[L1,λ,a]]. Then,
(1.1) possesses a leas wo coexis ence s a es o each µ∈(µ∗, σΩ
1[L2−c(x)θ[L1,λ,a]]).
P oo . Being he p oo a he simila o he p oo o Theo em 8.10, we a e only o ske ch
i . By symme y, i su ices o show Pa (ii).
Le (µ∗, u∗, ∗) be a coexis ence s a e o (1.1). Then, i is easily seen ha o each
µ∈(µ∗, σΩ
1[L2−c(x)θ[L1,λ,a]])
x= (u∗, ∗), x = (K1, K2),
is an o de ed sub-supe solu ion pai o (1.1) p o ided K1and K2a e su icien ly la ge
posi i e cons an s. Mo eo e , hanks o Lemma 6.2, i K1and K2a e su icien ly la ge,
hen any coexis ence s a e o (1.1) lies in he o de in e al [0, x]. The e o e, (1.1)
possesses a maximal coexis ence s a e wi hin he in e al [x, x], deno ed by (uµ, µ).
Thanks o P oposi ion 7.8 o [2], (uµ, µ) is weakly s able and so σΩ
1[Lµ]≥0 whe e Lµ
is he ope a o de ined by (8.5) wi h A(x) gi en by (8.3) and (u0, 0) = (uµ, µ).
I σΩ
1[Lµ]>0 he same a gumen o he p oo o Theo em 8.10 comple es he p oo
o Theo em 8.11.
I σΩ
1[Lµ] = 0 a guing as in he p oo o Theo em 8.10 we ind ha
µ(s)> µ ∀s∈(0, ε),
SYMBIOTIC SPECIES 51
and wo di e en si ua ions may a ise:
Case a. I µ(s)> µ o s∈(−ε, 0), hen he same a gumen o he p oo o Theo em
8.10 applies o comple e he p oo o his one.
Case b. I µ(s)< µ o s∈(−ε, 0), hen he e exis s s1<0 such ha µ0(s1)>0 and
hence,
i(Kµ(s1),(u(s1), (s1))) = 1 .
Now, se ing
ρ1:= k(uµ, µ)ke+δ, ρ2:= k(u(s1), (s1))ke+δ, ρ := k(u(s1), (s1))ke−δ.
yields
i(Kµ(s1), Pρ1 Pρ2)=0, i(Kµ(s1), Pρ1 Pρ) = 1 , i(Kµ, Pρ1 Pρ) = 1 ,
and he e o e,
i(Kµ,(uµ, µ)) = 1 .
This comple es he p oo . ¤
9. On he uniqueness o he coexis ence s a e. In his sec ion we gi e a unique-
ness esul in he case o small in e ac ion coe icien s. When he in e ac ion coe icien s
a e la ge we al eady know ha (1.1) exhibi s a supe linea cha ac e and so i s numbe
o coexis ence s a es migh a y d as ically when he geome y o he suppo domain
Ω changes, [8]. Ou main uniqueness esul is he ollowing.
Theo em 9.1. Assume ha (6.1),(6.8) and (8.1) a e sa is ied and ha o any coex-
is ence s a e (u0, 0)o (1.1)
µu0
0¶Mµ 0
u0¶M
<³a
b´Lµd
c¶L
.(9.1)
Then, (1.1) possesses a unique coexis ence coexis ence. Mo eo e , i is exponen ially
asymp o ically s able.
A e he p oo o his heo em we shall use Theo em 8.7 o ge some uppe es ima es
o he le hand side o (9.1), gi ing ise o e y simple easily compu able su icien
condi ions, in e ms o he se e al coe icien s in ol ed in he model se ing, o he
uniqueness o he coexis ence s a e.
P oo . Unde condi ions (6.1) and (6.8) we ha e uni o m a p io i bounds o he non-
nega i e solu ions o (1.1) and hence he ixed poin index in cones can be used as
in Sec ion 8.3. By P oposi ion 4.1 he semi- i ial posi i e solu ions (θ[L1,λ,a],0) and
(0, θ[L2,µ,d]) a e linea ly uns able, i hey exis , and a a he s anda d index compu a ion
shows ha each o hem has local index ze o (c . [23] o de ails). Mo eo e , he s a e
(0,0) has index ze o and he global index equals one. The e o e, by he p inciple
52 M. DELGADO, J. L´
OPEZ-G´
OMEZ AND A. SU´
AREZ
o linea ized s abili y, i su ices o show ha unde condi ion (9.1) any coexis ence
s a e is linea ly asymp o ically s able, since by Le ay-Schaude o mula any linea ly
asymp o ically s able solu ion has local index one. Le (u0, 0) be a coexis ence s a e
o (1.1). Then, he spec um o he linea iza ion o (1.1) a (u0, 0) is gi en by he
τ’s o which he ollowing p oblem has some solu ion (u, )∈W2,p
0(Ω) ×W2,p
0(Ω),
(u, )6= (0,0), p > N,
(L1+ 2au0−b 0−λ)u=bu0 +τu ,
(L2+ 2d 0−cu0−µ) =c 0u+τ . (9.2)
By Theo em 8.3 i we a e able o show ha he e exis u > 0 and > 0 such ha
(L1+ 2au0−b 0−λ)u > bu0 , (L2+ 2d 0−cu0−µ) > c 0u , (9.3)
hen he p incipal eigen alue o (9.2) will be posi i e and he e o e, he linea ized s a-
bili y o (u0, 0) will ollow om Theo em 8.3. Taking (u, ) = (αu0, β 0), whe e α > 0
and β > 0 ha e o be ound, (9.3) becomes in o
αau0> βb 0, βd 0> αcu0.(9.4)
Now, due o (9.1), i is a he clea ha he e exis α > 0 and β > 0 sa is ying (9.4).
This comple es he p oo . ¤
The ollowing esul p o ides us wi h a su icien condi ion o (9.1) o be hold.
P oposi ion 9.2. Assume L1=L2,b(x)>0and c(x)>0 o each x∈Ω,
σΩ
1[L1]>0, bMcM< aLdL, λ > σΩ
1[L1], µ > σΩ
1[L1],(9.5)
and
aMdM
16aLdL(aLdL−bMcM)2·(dLλ2+bMµ2)(aLµ2+cMλ2)
(λ−σΩ
1[L1])(µ−σΩ
1[L1]) ·Ãsup
Ω
ψ
ϕ!2
<1
bMcM
,(9.6)
whe e ϕ > 0is he p incipal eigen unc ion associa ed wi h σΩ
1[L1], no malized so ha
kϕkL∞(Ω) = 1 and ψ > 0is he unique solu ion o
L1ψ= 1 in Ω, ψ|∂Ω= 0 .
Then (1.1) has exac ly one coexis ence s a e.
P oo . We claim ha o each > 1 he couple (u , ) de ined by
u := (dLλ2+bMµ2)
4(aLdL−bMcM)ψ , := (aLµ2+cMλ2)
4(aLdL−bMcM)ψ ,
SYMBIOTIC SPECIES 53
is a s ic supe solu ion o (1.1). To p o e his i su ices o show ha
1≥ψ·[λ− (a(x)K1−b(x)K2)ψ],
1≥ψ·[µ− (d(x)K2−c(x)K1)ψ],(9.7)
whe e
K1=dLλ2+bMµ2
4(aLdL−bMcM), K2=aLµ2+cMλ2
4(aLdL−bMcM).
Since
sup
ξ≥0
(A−Bξ)ξ=A2
4B,
we ind ha o each ≥1,
ψ·[λ− (a(x)K1−b(x)K2)ψ]≤λ2
4 (a(x)K1−b(x)K2)≤λ2
4(aLK1−bMK2).
Simila ly,
ψ·[µ− (d(x)K2−c(x)K1)ψ]≤µ2
4(aLK2−bMK1).
Thus, he ollowing condi ions imply (9.7)
λ2= 4(aLK1−bMK2), µ2= 4(aLK2−bMK1).
Since hese condi ions a e sa is ied by he choice o K1and K2i sel , he claim abo e
ge shown.
Now, we need he ollowing gene alized e sion o he sweeping maximum p inciple
o [28], whose p oo is pos poned up o he end o he p oo o P oposi ion 9.2.
Lemma 9.3. Le x= (u, )∈W2,p
0(Ω) ×W2,p
0(Ω),p > N, be a solu ion o he p oblem
L1u= (x, u, )
L2 =g(x, u, )in Ω,
u= = 0 on ∂Ω,
whe e and ga e wo con inuous unc ions in xand o class C1in (u, ), inc easing
in , and ginc easing in u. Fo each ∈( 0, 1], le x = (u , )∈W2,p
0(Ω) ×W2,p
0(Ω)
be a s ic supe solu ion o his p oblem. Assume ha x is con inuous and s ic ly
inc easing in , ha x 1−xis s ongly posi i e, and ha ∂nx is con inuous in , whe e
ns ands o he ou wa d uni no mal o Ω. Then,
x≤x 0.

54 M. DELGADO, J. L´
OPEZ-G´
OMEZ AND A. SU´
AREZ
Thanks o Lemma 9.3, we ind ha
u0≤dLλ2+bMµ2
4(aLdL−bMcM)ψ , 0≤aLµ2+cMλ2
4(aLdL−bMcM)ψ , (9.8)
o any coexis ence s a e (u0, 0) o (1.1). Simila ly, i ollows om Lemma 3.2 ha
u0≥θ[L1,λ,a]≥λ−σΩ
1[L1]
aM
ϕ , 0≥θ[L1,µ,d]≥µ−σΩ
1[L1]
dM
ϕ . (9.9)
Finally, using (9.8) and (9.9), i is easily seen ha (9.6) implies (9.1). Theo em 9.1
comple es he p oo . ¤
P oo o Lemma 9.3. Le ∗deno e he in imum o he se o ∈( 0, 1) o which x−x
is s ongly posi i e. We claim ha ∗= 0. On he con a y, assume ha ∗> 0.
By ou assump ions i is a he clea ha he e exis s K > 0 such ha each o he
mappings
u→ (·, u, ) + Ku , →g(·, u, ) + K ,
is inc easing and K > −min{σΩ
1[L1], σΩ
1[L2]}. Since x ∗is a s ic supe solu ion o he
p oblem, some o i s componen s, say u ∗, sa is ies
(L1+K)(u ∗−u)> (·, u ∗, ∗) + Ku ∗− (·, u, )−Ku > 0.
Thus, he s ong maximum p inciple implies ha u ∗−uis s ongly posi i e. This
con adic s he minimali y o ∗and comple es he p oo . ¤
No e ha , hanks o he s ong maximum p inciple, ϕand ψa e s ongly posi i e
and hence, supΩ
ψ
ϕis well de ined.
The es ima es gi en by he ollowing esul will be used o ind ou ano he su icien
condi ion o (9.1).
Lemma 9.4. Assume L1=L2,b(x)>0and c(x)>0 o each x∈Ω, and
bMcM< aLdL, λ ≥µ > σΩ
1[L1].
Then, o any coexis ence s a e (u, )o (1.1) he ollowing es ima es hold
M1θ[L1,µ,d]≤u≤N1θ[L1,λ,a],(9.10)
M2θ[L1,µ,d]≤ ≤N2θ[L1,λ,a],(9.11)
whe e
N1=aM(dL+bM)
aLdL−cMbM
, N2=aM(aL+cM)
aLdL−cMbM
,
M1= max ½dL(bL+dM)
aMdM−cLbL
,(bL+dL)[dM(aM+cM)−cLdL]
aM[dM(aM+cM)−cL(bL+dL)]¾,
SYMBIOTIC SPECIES 55
M2= max ½dL(cL+aM)
aMdM−cLbL
,dM(aM+cM)
dM(aM+cM)−cL(bL+dL)¾.
P oo . Since
N1aL−bMN2=aM, N2dL−N1cM=aM,
o each ≥1 we ha e ha
(N1aL−N2bM)−aM≥0, (N2dL−N1cM)−aM≥0.(9.12)
Now, hanks o (9.12) i is easily seen ha o each > 1 he couple (u , ) de ined by
(u , ) := (N1θ[L1,λ,a], N2θ[L1,λ,a])
is a s ic supe solu ion o (1.1). The e o e, hanks o Lemma 9.3, he uppe es ima es
in (9.10) and (9.11) ge shown.
Now, in o de o p o e he alidi y o he lowe es ima es in (9.10), (9.11) we will
adap a de ice coming om [17]. A ei e a i e applica ion o Lemma 3.2 shows ha
αnθ[L1,µ,d]≤u , βnθ[L1,µ,d]≤ , (9.13)
o each n≥1, whe e
αn=dL+bLβn−1
aM
, βn=dL+cLαn−1
dM
, α0=dL/aM, β0= 1 .
Thus, passing o he limi as n→ ∞ yields
αθ[L1,µ,d]≤u , βθ[L1,µ,d]≤ ,
whe e
α=dL(bL+dM)
aMdM−bLcL
, β =dL(cL+aM)
aMdM−bLcL
.
This p o ides us wi h hal o he lowe es ima es in (9.10), (9.11). Now, i ollows om
Lemma 7.2 (ii) ha /K ≥θ[L1,µ,d], whe e
K=dM(cM+aM)
dM(aM+cM)−cL(bL+dL).
Thus,
L1=λu −a(x)u2+b(x)u ≥µu −a(x)u2+bLKθ[L1,µ,d]u
and hence, uis a supe solu ion o
L1w= (µ+bLKθ[L1,µ,d])w−a(x)w2in Ω ,
w= 0 on ∂Ω.(9.14)
56 M. DELGADO, J. L´
OPEZ-G´
OMEZ AND A. SU´
AREZ
The e o e, Theo em 3.1 implies
θ[L1−bLKθ[L1,µ,d],µ,a]≤u .
Finally, a u he applica ion o Lemma 3.2 shows ha
Bθ[L1,µ,d]≤θ[L1−bLKθ[L1,µ,d],µ,a],
whe e
B=(bL+dL)[dM(aM+cM)−cLdL]
aM[dM(aM+cM)−cL(bL+dL)] .
This comple es he p oo . No e ha Kand Ba e posi i e cons an s. ¤
Now, as an immedia e consequence om Theo em 9.1 and Lemma 9.4 we ob ain he
ollowing esul .
Co olla y 9.5. Assume L1=L2,b(x)>0and c(x)>0 o each x∈Ω,
bMcM< aLdL, λ ≥µ > σΩ
1[L1],
and
N1
M2·N2
M1Ãsup
Ω
θ[L1,λ,a]
θ[L1,µ,d]!2
<aLdL
bMcM
.(9.15)
Then, (1.1) possesses a unique coexis ence s a e.
No e ha since θ[L1,λ,a]and θ[L1,µ,d]a e s ongly posi i e, supΩ
θ[L1,λ,a]
θ[L1,µ,d]is well de ined.
Rema k 9.6. (i) I a,b,cand da e assumed o be cons an , hen
M1=d(b+d)
ad −cb , M2=d(c+a)
ad −cb ,
al hough in case a=b=c= 1 he e a e choices o d(x) o which some o hese ela ions
ails.
(ii) I a,b,cand da e cons an , hen (9.15) becomes in o he condi ion ound in
Theo em 3.3 o [17].
(iii) As a consequence om P oposi ion 9.2 and Co olla y 9.5, i ollows ha i one
o he in e ac ion coe icien s (bo c) is small, hen (1.1) possesses a unique coexis ence
s a e. Fo some special classes o domains and di e en ial ope a o s, how small should
be bo c o ha e uniqueness can be es ima ed in e ms o he se e al coe icien s o he
model. Fo ins ance, i Ω = (0, π), L1=L2=−d2
dx2and a=d= 1, hen σΩ
1[L1] = 1,
ϕ(x) = sin(x), ψ(x) = x(π−x)/2, supΩ
ψ
ϕ=π/2 and he es ima e (9.15) becomes in o
R(λ, µ) := sup
Ω
θ[L1,λ,1]
θ[L1,µ,1]
<1
√bc .(9.16)
SYMBIOTIC SPECIES 57
Some explici es ima es o R(λ, µ) we e ound in [17] and [1]. Namely, in [17] i was
shown ha
R2(λ, µ)≤λ3
(µ−1)2.(9.17)
The e o e, hanks o Co olla y 9.5, (1.1) possesses a unique coexis ence s a e p o ided
bc < (µ−1)2
λ3.(9.18)
(i ) In many cases P oposi ion 9.2 is sha pe han Co olla y 9.5. Indeed, in he
p e ious example (9.6) becomes in o
bc < 64
π2·(λ−1)(µ−1)(1 −bc)2
(λ2+bµ2)(µ2+cλ2).(9.19)
Thus, i λ= 2, µ= 1.5 and c= 1, (9.18) becomes in o b < 1/32 ≃0.031, while (9.19)
becomes in o b < b0wi h b0≃0.099. The e o e, in his case (9.19) is sha pe han
(9.18).
Unde he assump ions o Theo em 9.1, he p oblem o he global a ac i i y o
he coexis ence s a e wi h espec o he cone o posi i e unc ions in bo h componen s
is e y di icul o handle wi h. This is in s ong con as wi h he compe ing species
coun e pa o (1.1), whe e due o he comp essi i y o he model (c [14]) he uniqueness
o a s able coexis ence s a e implies i s global a ac i i y as a esul om he abs ac
heo y o [9]. Ne e heless, he p esence o uni o m a p io i bounds in he con ex o
Theo em 9.1 allows us o apply he ollowing esul o [15] o he pa abolic sys em
associa ed wi h (1.1).
Theo em 9.7. Assume ha Tis a s ongly posi i e mono one con inuous dynamical
sys em on Xwhe e he cone Khas non-emp y in e io and Xis sepa able. Mo eo e ,
assume ha O(x)( he posi i e semi-o bi o x) is compac o each x∈X. Then,
he e exis s a dense subse Ao Xsuch ha i x∈A, hen ω(x)( he ω-limi o x), is
con ained in he se o s a iona y poin s.
Using his esul we ob ain he ollowing one.
Theo em 9.8. Assume ha bMcM< aLdL,λ>σΩ
1[L1],µ>σΩ
1[L2],b(x)>0,
c(x)>0, o each x∈Ω, and ha (1.1) possesses a unique coexis ence s a e, say
(uc, c). Conside he ollowing pa abolic eac ion di usion p oblem
∂ u+L1u=λu −au2+bu ,
∂ +L2 =µ −d 2+cu , in Ω×(0,∞),
u|∂Ω= |∂Ω= 0 , > 0,
u(x, 0) = u0(x), (x, 0) = 0(x), x ∈Ω,
64 M. DELGADO, J. L´
OPEZ-G´
OMEZ AND A. SU´
AREZ
We should poin ou ha all he p e ious in o ma ion is o local na u e, i.e. i p o ides
us wi h he bi u ca ion di ec ions o coexis ence s a es om he semi- i ial s a es o
alues o he pa ame e s close o he co-dimension wo singula i y (σΩ
1[L1], σΩ
1[L2]).
Being he p oblem o inding ou global in o ma ion abou he na u e o hese local
bi u ca ions e y di icul o handle wi h in ou gene al se ing, in he nex sec ion we
will es ic ou sel es o he conside he e y special case when L1=L2=−∆ and
all he coe icien s a e cons an . In pa icula , i will be shown ha he e a e anges o
he pa ame e s o which he e is a change o he bi u ca ion di ec ion o coexis ence
s a es p o ided ad−bc > 0 is su icien ly small. This will p o ide us wi h some su icien
condi ions so ha he model exhibi s a leas wo coexis ence s a es acco dingly o he
mul iplici y esul s o Sec ion 8.
I A= 1, hen Theo em 10.1 can no be applied and he complexi y o he bi u ca ion
diag ams inc ease. In his case, Theo em 5.1 (ii) o [11] gi es he ollowing esul .
Theo em 10.3. Assume A= 1, and se
ε=sign a0, c =−a2
3a2c0|a0|−1,
whe e
a0=−1
2
a4
a3
b2+b3−1
2
a2
a3
b5+a2
a4
b6+1
2
a2a4
a3
(d1+d3)−1
2a2(d2+d4),
c0=a4
a2
b1−1
2
a3
a2
b2+b4−1
2
a3
a4
b5−1
2a3(d1+d3) + 1
2
a2
3
a4
(d2−d4).
Then, i a0c0((c0)2−(a0)2)6= 0, is K-equi alen o
à (λ− +s−εs2)
s(µ+ −s−c 2)!.
Mo eo e , he uni e sal un olding o is gi en by
µ (λ− + (1 + β)s−εs2)
s(µ+ −s−c 2)¶(10.8)
and cis a modal pa ame e . He e, β≃0is an un olding pa ame e .
F om (10.8), he bi u ca ion di ec ions o coexis ence s a es can be e y easily ound
ou . In ou p esen si ua ion, he signs o he p qs−q psdepend on he pa ame e ,
as shown by he ollowing iden i ies
(p qs−q ps)(Λ1,0( )) = −β+2(1+β)c +... , (p qs−q ps)(Λ0,1( )) = −β+2ε +... .
No ice ha since Λ1,0(0) = Λ0,1(0) = 0, when g ows Λ1,0( ) and Λ0,1( ) sepa a e om
(σΩ
1[L1], σΩ
1[L2]).

SYMBIOTIC SPECIES 65
The lis bellow p o ides us wi h all he bi u ca ion di ec ions as sg ows om ze o.
Wi hou los o gene ali y, we can assume ha ε= 1.
1. Bi u ca ion di ec ions along Λ0,1
1.1- I β > 0, hen o alues o he pa ame e s su icien ly close o (σΩ
1[L1], σΩ
1[L2])
he bi u ca ion o coexis ence s a es is subc i ical, up o some alue o he pa ame e
whe e i becomes in o supe c i ical.
1.2- I β < 0, hen he bi u ca ion is always supe c i ical.
2. Bi u ca ion di ec ions along Λ1,0
2.1- I c > 0 and β > 0, hen he si ua ion desc ibed in case 1.1 occu s.
2.2- I c > 0 and β < 0, hen he bi u ca ion di ec ion is supe c i ical.
2.3- c < 0 and β > 0, hen he bi u ca ion di ec ion is subc i ical.
2.4- I c < 0 and β < 0, hen o alues o he pa ame e s su icien ly close o
(σΩ
1[L1], σΩ
1[L2]) he bi u ca ion is supe c i ical, while a e some c i ical alue becomes
subc i ical.
We should poin ou ha , due o he symme y o he p oblem, i L1=L2=−∆
and a0=c0= 0, hen is much mo e degene a e han (10.8). To ea hese degene a e
si ua ions we e e o he Appendix o [10].
11. The special case L1=L2=−∆wi h cons an coe icien s. Th oughou his
sec ion we assume ha L1=L2=−∆ and ha a,b,cand da e cons an . A e a
change o a iables we can assume ha
a=d= 1 .
In he sequel we use he no a ion
σ1[q] := σΩ
1[−∆ + q], σ1:= σ1[0] , θγ:= θ[−∆,γ,1] ,
and ex end he de ini ion o θγ aking θγ:= 0 o γ≤σ1. As an immedia e consequence
om he esul s in he p e ious sec ions we ob ain he ollowing global heo em, which
is a subs an ial imp o emen o all he p e ious esul s in he e e ences.
Theo em 11.1. (i) Assume bc < 1. Then, he ollowing asse ions a e ue:
(i.1) I any o he semi- i ial posi i e solu ions is linea ly uns able, hen (1.1) possesses
a coexis ence s a e. I in addi ion λ > σ1,µ > σ1, hen he e exis s I0>0such
ha i ei he b < I0o c < I0, hen he coexis ence s a e is unique and exponen ially
asymp o ically s able.
(i.2) I o (λ, µ)=(λ0, µ0)some o he semi- i ial posi i e solu ions is linea ly s able
and (1.1) possesses a coexis ence s a e, hen i possesses a coexis ence s a e o each
(λ, µ)sa is ying λ≥λ0,µ≥µ0, and a leas wo coexis ence s a es i λ > λ0,µ > µ0
and some o he semi- i ial posi i e solu ions is linea ly s able.
66 M. DELGADO, J. L´
OPEZ-G´
OMEZ AND A. SU´
AREZ
(i.3) Fo each λ∈R, he e exis s µex (λ)∈Rsuch ha (1.1) does no admi a coexis-
ence s a e i µ≤µex (λ). Simila ly, o each µ∈R, he e exis s λex (µ)∈Rsuch ha
(1.1) does no admi a coexis ence s a e i λ≤λex (µ). Mo eo e , hanks o Lemma
6.2,
µex (λ)≥(1 −bc)σ1−cλ , λex (µ)≥(1 −bc)σ1−bµ . (11.1)
(ii) Assume bc > 1. Then, he ollowing asse ions a e ue:
(ii.1) I N≤5and some o he semi- i ial posi i e solu ions is linea ly s able, hen
(1.1) possesses a coexis ence s a e.
(ii.2) I N≤5and he e exis s (λ, µ) = (λ0, µ0) o which (1.1) possesses a coexis ence
s a e being any o he semi- i ial s a es linea ly uns able, hen (1.1) possesses a coex-
is ence s a e o each (λ, µ)sa is ying λ≤λ0and µ≤µ0, and a leas wo coexis ence
s a es i λ < λ0and µ < µ0and any o he semi- i ial s a es is linea ly uns able.
(ii.3) Fo each λ∈R, he e exis s µex (λ)∈Rsuch ha (1.1) does no admi a coexis-
ence s a e i µ≥µex (λ). Simila ly, o each µ∈R, he e exis s λex (µ)∈Rsuch ha
(1.1) does no admi a coexis ence s a e i λ≥λex (µ).
The i s goal o his sec ion is inding ou sha pe es ima es han (11.1) o he alues
o λex (µ) and µex (λ) in he case bc < 1. Ou main esul in his di ec ion eads as
ollows:
Theo em 11.2. Assume bc < 1and
λ > σ1, λ ≥µ > σ1[−c1 + b
1−bcθλ].(11.2)
Then,
u≤1 + b
1−bcθλ, ≤θ[−∆−c1+b
1−bc θλ,µ,1] ,(11.3)
o any coexis ence s a e (u, )o (1.1). The e o e, i λ > σ1and
µ≤max{σ1[−c1 + b
1−bcθλ], σ1(1 −bc)−cλ }(11.4)
hen (1.1) does no admi a coexis ence s a e. By symme y, he same esul holds i
µ > σ1and
λ≤max{σ1[−b1 + c
1−bcθµ], σ1(1 −bc)−bµ }.
P oo . Thanks o Lemma 7.2, (1+b) ≤(1+c)uand hence, we ind om he u-equa ion
o he sys em ha
−∆u≤λu −1−bc
1 + bu2.
SYMBIOTIC SPECIES 67
Thus, Lemma 3.2 implies he i s uppe es ima e o (11.3). Subs i u ing his es ima e
in o he -equa ion o he sys em gi es
(−∆−c1 + b
1−bc θλ) ≤µ − 2,
and Lemma 3.2 comple es he p oo o (11.3). The emaining asse ions ollow eadily
om Theo em 3.1 and Theo em 11.1 (i.3). ¤
Rema k 11.3. The cu e de ined by he igh hand side o (11.4) mee s (σ1, σ1) a he
alue λ=σ1, since limλ↓σ1θλ= 0 and hence,
lim
λ↓σ1
max{σ1[−c1 + b
1−bcθλ], σ1(1 −bc)−cλ }= lim
λ↓σ1
σ1[−c1 + b
1−bcθλ] = σ1,
hanks o he con inuous dependence o he p incipal eigen alue wi h espec o he
po en ial. The e o e, he es ima e o he ex inc ion egion gi en by (11.4) is op imal o
alues o λ≃σ1.
Mo eo e , (11.4) is also op imal o alues o λ a ying on compac subin e als o
[σ1,∞) p o ided bis su icien ly small, as he ollowing esul shows.
Theo em 11.4. Assume bc < 1,λ > σ1and µ < σ1[−cθλ]. Then, he e exis s b0=
b(λ)>0such ha (1.1) does no admi a coexis ence s a e i b∈[0, b0]. Mo eo e , b(λ)
a ies con inously wi h λ.
P oo . The unc ion
h(b) := −c1 + b
1−bc ,
is dec easing and i sa is ies
h(0) = −c , lim
b↑c−1h(b) = −∞.
Thus, he e exis s a unique b0=b(λ)>0 such ha
µ=σ1[−c1 + b0
1−b0cθλ]< σ1[−cθλ].
The e o e, o b∈[0, b0] we ha e ha
µ≤σ1[−c1 + b
1−bc θλ]≤σ1[−cθλ]
and Theo em 11.2 comple es he p oo . ¤
Rema k 11.5. Thanks o he es ima e (4.10) in he p oo o Theo em 4.1 in [25], we ind
ha
σ1[−c1 + b
1−bcθλ]≤σ1−c1 + b
1−bc (λ−σ1)
68 M. DELGADO, J. L´
OPEZ-G´
OMEZ AND A. SU´
AREZ
and he e o e, he ollowing es ima e o µex (λ) is ob ained
µex (λ)≥(σ1−c1+b
1−bc (λ−σ1) i λ≤σ1b(2−bc)+1
b(c+1) ,
σ1(1 −bc)−cλ i λ > σ1b(2−bc)+1
b(c+1) .
This es ima e p o ides us wi h some e y eadily compu able su icen condi ion in
e ms o he se e al coe icien s in ol ed in he model se ing o he ex inc ion o he
species .
In Figu e 4 we ha e ep esen ed he cu e o change o s abili y o (θλ,0) oge he
wi h he bounda y o he ex inc ion egion gi en by he es ima e (11.4); o alues o
(λ, µ) in he b igh g ey egion he model possesses a coexis ence s a e, while o he
alues o (λ, µ) in he da ke egion he species is d i en o ex inc ion by u.
λ
µ
µ=
(σ
F
1
(λ)
,σ 1)
Figu e 4: The coexis ence and ex inc ion egions.
In he nex esul we comple e he local analysis o Sec ion 10 by gi ing some su icien
condi ions o comple ely asce aining he bi u ca ion di ec ions o coexis ence s a es in
he case bc > 1.
SYMBIOTIC SPECIES 69
Theo em 11.6. Assume bc > 1,bc ≥2 + cand ix λ>σ1. Then he bi u ca ion
di ec ion o coexis ence s a es om (µ, u, )=(σ1[−cθλ], θλ,0) is subc i ical. By sym-
me y, i bc > 1,bc ≥2 + band we ix µ>σ1, hen he bi u ca ion di ec ion om
(λ, u, ) = (σ1[−bθµ],0, θµ)is subc i ical.
P oo . Le (µ(s), u(s), (s)) deno e he local cu e o coexis ence s a es emana ing om
(θλ,0) a µ=σ1[−cθλ]. The main heo em o [7] gua an ees ha µ(s) is eal analy ic
in sand hence i possesses an expansion o he o m
µ(s) = σ1[−cθλ] + sµ1(λ) + O(s2),as s→0,
o some µ1(λ)∈R. A a he s anda d calcula ion shows ha (c . [6] and [10] o
de ails)
µ1(λ) = (2 + c)−1[(2 + c−bc)ZΩ
ϕ3
λ−bc(λ−σ1[−cθλ]) ZΩ
ϕ2
λR(λ)ϕλ],(11.5)
whe e R(λ) := (−∆+2θλ−λ)−1and ϕλ>0 is he p incipal eigen unc ion associa ed
wi h σ1[−cθλ] no malized so ha kϕλk2= 1. This comple es he p oo . ¤
Modulo he change o band cby −band −c, espec i ely, he o mula (5.2) o [10]
p o ides us wi h he sign o µ1(λ) o λ≃σ1.
Lemma 11.7. (i) I λis su icien ly close o σ1, hen
sign µ1(λ) = sign (1 −bc).
(ii) Simila ly, o µ≃σ1,
sign λ1(µ) = sign (1 −bc),
whe e λ1(µ) = dλ
ds |s=0. He e, λ(s)s ands o he λ-componen o he cu e o coexis ence
s a es emana ing om (λ, u, ) = (σ1[−bθµ],0, θµ), whose exis ence is gua an eed by
Theo em 5.1.
We now show how change he bi u ca ion di ec ions o coexis ence s a es along he
semi- i ial b anches as bc g ows om he c i ical alue 1, so comple ing he esul s o
Sec ion 10. Fo his we will use he local bi u ca ion analysis al eady done in Sec ion 10.
In e exchanging he oles o band cin [10] by −bby −che e, we ob ain he bi u ca ion
equa ion
λ − p( , s, λ, µ, b, c) = 0 , µs −sq( , s, λ, µ, b, c) = 0 ,(11.6)
whe e q( , s, λ, µ, b, c) = p(s, , µ, λ, c, b) and
p( , s, λ, µ, b, c) = M( −bs) + N[2 2−b(3 −c) s −b(1 −b)s2]
+K{5 3−b(c2−4c+ 10) 2s−3b[(1 −b)(1 −c)−b] s2
−b(b2−2b+ 2)s3}
+L[2λ 2−b(3λ−cµ) s −b(µ−bλ)s2]
+O(4,( , s, λ, µ)) ,

70 M. DELGADO, J. L´
OPEZ-G´
OMEZ AND A. SU´
AREZ
whe e M,N,K,La e he cons an s de ined by (3.6) in [10]. We should poin ou ha i
bc = 1, hen he cons an s a0and c0o he s a emen o Theo em 10.2 equal ze o, and so
Theo em 10.2 does no co e his case. This is why o analyze he change o c i icali y
o he local bi u ca ions om he semi- i ial b anches hi d o de e ms a e needed.
Ou main esul in his di ec ion is he ollowing, whe e he no a ions in oduced in
Sec ion 10 a e kep .
Theo em 11.8. I bc −1>0is su icien ly small, hen he e exis s a unique change o
c i icali y in a neighbo hood o he o igin along each o he cu es Mλand Mµ.
P oo . A e some s igh o wa d manipula ions, we ind ha
λ1( ) = M + 2N 2+ (5K+ 2LM) 3+O( 4),(11.7a)
µ1( ) = −Mc −Nc(1 −c) 2−(Kc(c2−2c+ 2) + LMc(1 + c2)) 3+O( 4).(11.7b)
Thus, se ing
Jac1( ) = (p qs−psq )( , 0, λ1( ), µ1( )) ,
and subs i u ing (11.7) in i gi es
Jac1( ) = εM2+ε(4 −3c)NM + [2(c+ 1)2(KM −N2) + εFc] 2+O( 3),
whe e
ε:= 1 −bc , Fc=M2L(3c2+ 4) + KM(4c2−7c+ 13) + N2(2c2−8c+ 2) .
Making he change o a iables
ε=−τ2, s =s0τ ,
and se ing
Jac1(τ, s0) := Jac1(−τ2, s0τ)
τ2, P := 2(c+ 1)2(KM −N2),
i is easily seen ha
Jac1(τ, s0) = −M2−(4 −3c)NMs0τ+Ps2
0−τ2s2
0Fc+O(s3
0τ),
We al eady know ha P > 0 (c . [10], pg. 109). Mo eo e , we ha e ha
Jac1(0,M
√P) = 0 , Ds0Jac1(0,M
√P) = 2√PM 6= 0 .
Thus, hanks o he implici unc ion heo em, he e exis s a unique unc ion s0such
ha o each τ≃0
s0(0) = M(P)−1/2,Jac1(τ, s0(τ)) = 0 .
Hence o h,
Jac1(−τ2, s0(τ)τ) = 0 .
The e o e, he e exis s a unique (ε)>0 such ha
Jac1( (ε)) = 0 .
By symme y, he emaining asse ions ge shown. This comple es he p oo . ¤
SYMBIOTIC SPECIES 71
Some u he discussion. We now summa ize he in o ma ion gi en by he esul s in
he las wo sec ions. Fo his, i is con enien ega ding band cas he main pa ame e s
o he model. Mo e p ecisely, we will ix c > 0 and a y b. Thanks o Lemma 10.2, i
b < c−1, hen he bi u ca ion di ec ions o coexis ence s a es a e supe c i ical. Thanks
o Theo em 11.8, he e exis s ε0=ε0(c)>0 such ha i c−1<b<(1 + ε0)c−1 hen
he bi u ca ion di ec ions a e subc i ical o (λ, µ) close o (σ1, σ1), in ac his holds
in a √bc −1-neighbo hood o (σ1, σ1), while hey become supe c i ical ou side his
neighbo hood, wi hin ano he sligh ly la ge neighbo hood o (σ1, σ1). Now, since he
cu es bc = 2+band bc = 2+cin he s a emen o Theo em 11.6 mee a (b, c) = (2,2),
changing hei ela i e posi ions as cac osses 2, wo di e en cases mus be conside ed.
I c < 2, hen we ind om Theo em 11.6 ha (1 + ε0)c−1<1 + 2
c, since o bc ≥c+ 2
all bi u ca ion di ec ions om (θλ,0) became subc i ical. I c < 1, hen ou esul s
do no p o ide us wi h any u he global in o ma ion abou he bi u ca ion di ec ions
along (0, θµ), while in case 1 <c<2 i ollows om Theo em 11.6 ha i binc eases
up o ac ossing some c i ical alue, necessa ily less han 2
c−1, hen all bi u ca ions o
coexis ence s a es om (0, θµ) will change o subc i ical ei he . In case c > 2 hese
global changes in he na u e o he bi u ca ions occu in he con e se o de . Now, any
bi u ca ion di ec ion om (θλ,0) is subc i ical i b > 2
c+1 and mo eo e all bi u ca ion
di ec ions om any o he semi- i ial s a es a e subc i ical i b > 1 + 2
c.
c
b
bc=1
bc=2+b
bc=2+c
G
L
G
λ
Gµ
Figu e 5: Va ying band c.
In Figu e 5 we ha e summa ized all he p e ious in o ma ion. The i s quad an is
di ided in o ou egions. The b igh g ey egion s ands o bc < 1, whe e we only ha e
local in o ma ion; he black egion, which is a hin s eep abo e bc > 1, whe e we know
72 M. DELGADO, J. L´
OPEZ-G´
OMEZ AND A. SU´
AREZ
ha he local change o c i icali y occu s; he egions Gλand Gµ, in be ween bc = 2 + c
and bc = 2+b, whe e we know ha he bi u ca ion di ec ion om one o he semi- i ial
b anches, espec i ely (θλ,0) and (0, θµ), is always subc i ical bu no global in o ma ion
abou he na u e o he bi u ca ion along he emaining semi- i ial b anch is a ailable;
in he egion G, hanks o Theo em 11.6 all bi u ca ion di ec ions a e subc i ical, and
inally he egion L, whe e only local in o ma ion is supplied by ou analysis. By he
con inuous dependence o he bi u ca ion di ec ions wi h espec o (λ, µ, b, c), i we
mo e away om L owa ds Gλ∪Gµ(o he egion G), any poin o change o c i icali y
on any o he semi- i ial b anches should a y along his b anch up o ei he mee wi h
ano he poin o change o c i icali y o g ow up o in ini y. In he i s case, bo h poin s
o change o c i icali y sh ink a he mee ing alue and hen dismiss. To comple e ou
discussion, in Figu e 6 we ha e ep esen ed a ypical bi u ca ion diag am o a alue o
(b, c) lying he black a ea o Figu e 5; a alue o (b, c) whe e he poin s o change o
c i icali y a e s ill close o he co-dimension wo singula i y (σ1, σ1).
λ
µ
µ= F(λ)
(σ1,σ1)
Figu e 6: Local bi u ca ion diag ams along he cu e o change o s abili y.
Acknowledgemen s. The au ho s hank o DGICYT o Spain o esea ch suppo
unde g an s DGICYT PB93-0465, DGICYT PB95-1242 and DGES PB96-0621. A.
Su´a ez also hanks o C´ama a Fonda ion o a Resea ch ellowship du ing he p epa a-
ion o his wo k.
SYMBIOTIC SPECIES 73
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