NON-NEGATIVE SOLUTIONS FOR A
HETEROGENEOUS DEGENERATE
COMPETITION MODEL
ANTONIO SU´
AREZ1
(Ma ch 30, 2004)
Abs ac
This pape deals wi h he exis ence, uniqueness and quali a i e p ope ies
o nonnega i e and non i ial solu ions o a spa ially he e ogeneous Lo ka-
Vol e a compe i ion model wi h nonlinea di usion. We gi e condi ions in
e ms o he coe icien s in ol ed in he se ing o he p oblem which assu e he
exis ence o nonnega i e solu ions as well as uniqueness o posi i e solu ion. In
o de o ob ain he esul s we employ mono onici y me hods, singula spec al
heo y and a ixed poin index.
Sho i le: Degene a e compe i ion p oblem
1. In oduc ion
In his wo k we a e mainly conce ned wi h he exis ence and uniqueness
o nonnega i e solu ions o he p oblem





L1(wm) = w(λ−a(x)w−b(x)z) in Ω,
L2(zn) = z(µ−d(x)z−c(x)w) in Ω,
w=z= 0 on ∂Ω,
(1)
whe e Ω is a bounded domain o IRNwi h egula bounda y ∂Ω, 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
ak
ij(x)DiDj+
N
X
i=1
bk
i(x)Dik= 1,2,(2)
1Dp o. Ecuaciones Di e enciales y An´alisis Num´e ico, Fac. Ma em´a icas, C/ Ta ia s/n,
C.P. 41012, Uni . Se illa, Spain, e-mail: sua ez@nume .us.es
c
°Aus alian Ma hema ical Socie y 0, Se ial- ee code 0334-2700/0
1
wi h ak
ij, bk
i∈C1(Ω); m, n > 1; λ, µ ∈IR and a, b, c, d ∈C1(Ω) nonnega i e
and non i ial.
P oblem (1) p o ides us wi h he s eady-s a e solu ions o a ela ed
e olu iona y p oblem, which models he beha iou o wo compe ing species,
wi h popula ions densi ies w(x) and z(x), inhabi ing Ω. We e e o [14] o
he meaning o each coe icien and de ails abou he model.
When m=n= 1 (linea di usion), (1) has been ex ensi ely s udied in
he ecen yea s. In he case ha a, b, c and da e s ic ly posi i e unc ions,
see o example [6], [7], [8], [10], [12], [13], [19], [20], [21], [25], [28], [32]
and he e e ences he ein. When band/o c anish in a domain o Ω ( ha
means ha , o ins ance, zdoes no in e ac wi h win he se B0:= {x∈
Ω : b(x)=0}); p oblem (1) was s udied in [22], [26] and [28]. And inally,
ecen ly he case a anishes in a pa o Ω bu all o he coe icien s unc ions
a e s ic ly posi i e o e Ω has been analysed in [18] and [27], whe e essen ial
quali a i e changes occu . Obse e ha in his case posi i e cons an s a e
no supe solu ions o (1) and, in ac , i is shown ha he a p io i bounds
a e los o some alues o λand µappea ing a new kind o posi i e solu ions
(which a e in ini e o e a egion o Ω and ini e on he es o Ω) ha go e n
he beha iou o a ela ed e olu iona y p oblem.
Howe e , model (1) is less known when m, n > 1, and i has been only
analysed unde mo e es ic i e hypo heses, wi h cons an coe icien s (ho-
mogeneous en i onmen al case) in [14] and when aand da e s ic ly posi-
i e in [9] and [31], all o hem wi h L1=L2=−∆. These new pa ame e s
(m, n) we e in oduced in [23] and [29] by desc ibing he dynamics o biolog-
ical popula ion whose mobili y depends upon hei densi y. In his con ex ,
i means ha he di usion, he a e o mo emen o he species om high
densi y egions o low ones, is slowe han in he linea case, gi ing mo e
ealis ic esul s. Ma hema ically, his has mainly h ee consequences which
dis inguish his sys em om he one wi h m=n= 1: he s ong maximum
p inciple does no apply (and so, unlike he linea case, he e can exis non-
nega i e and non i ial solu ions which a e no posi i e in all Ω), a-p io i
bounds o all he solu ions o (1) and o all he alues o λand µ, e en
when ao d anishes, exis and ha he linea ized me hod canno be applied
di ec ly.
In o de o s udy (1) we make he app op ia e change o a iables wm=u
and zn= , which ans o ms (1) in o





L1u=u1/m(λ−a(x)u1/m −b(x) 1/n) in Ω,
L2 = 1/n(µ−d(x) 1/n −c(x)u1/m) in Ω,
u= = 0 on ∂Ω.
(3)
Since only nonnega i e solu ions ha e physical in e es , he e a e ou ypes
o solu ions: he i ial one, he semi i ial solu ions (u, 0) and (0, ), hose
wi h bo h componen s s ic ly posi i e, he coexis ence s a es, and hose
2
whe e a leas one componen could anish in a pa o Ω, he semicoexis-
ence s a es. Obse e ha a semicoexis ence s a e could be a coexis ence
one (see P oposi ion 3.3). Some imes, we a e able o p o e ha a semicoex-
is ence s a e anishes in a egion o Ω (see Theo em 3.4), and so i is no a
coexis ence s a e.
Now we desc ibe he pa s o his wo k s a ing hei main esul s. Ob-
se e ha he semi i ial solu ions sa is y he ollowing equa ion, he eason
o ou s udy in Sec ion 2,
(Lw = (x)w1/ −g(x)w2/ in Ω,
w= 0 on ∂Ω,(4)
whe e Lis an ope a o o he o m (2), , g ∈C1(Ω) wi h g≥0, g6≡ 0,
can change sign and =mo n. Al hough he semi i ial solu ions gi e
≡λ(o µ) and so cons an , i will be e y use ul o s udy (4) when
changes sign. This equa ion has been p e iously s udied in [3], [14], [15], [24]
and [30] assuming mo e es ic ions in he da a o (4). We collec he main
esul s o hese wo ks, and as a consequence we ob ain ha he semi i ial
solu ion (u, 0) = ( esp. (0, )) exis s and i is unique i , and only i , λ > 0
( esp. µ > 0).
Then, we s udy he exis ence o dead co es (see [17]) o he solu ions o (4).
Gi en a solu ion wo (4); we call he se Ω0:= {x∈Ω : w(x) = 0}, i his
is nonemp y, a dead co e o w. We demons a e a esul which assu es he
exis ence o a dead co e o any nonnega i e solu ion o (4) unde sui able
hypo heses (see Theo em 2.4). A di ec consequence o ou esul is ha any
nonnega i e solu ion o (4) has dead co e i he maximum o is small. To
ou knowledge, he abo e esul s conce ning o he exis ence o dead co e
ha e been ob ained when L=−∆, see [3], [14], [17] and [30], wi h hei
p oo s being based on he adial p ope ies o he Laplacian. In his way
ou esul gene alises p e ious ones.
In Sec ion 3 we ca y ou an analysis o he exis ence o semicoexis ence,
coexis ence s a es and dead co es o he sys em (3). Using he esul s o
Sec ion 2 and mono onici y me hods we ob ain esul s which can be sum-
ma ized as ollows: ake λ∈IR,
•Assume λ≤0: i µ∈(−∞,0] only he i ial solu ion exis s, i
µ∈(0,∞) only he i ial and he semi i ial solu ions (0, ) exis ;
•Assume λ > 0: he e exis posi i e alues µ∗(λ), µ∗(λ), µ1(λ), µ2(λ)
wi h
µ1(λ)<min{µ∗(λ), µ∗(λ)}and µ2(λ)>max{µ∗(λ), µ∗(λ)}
such ha
–I µ∈(−∞,0] only he i ial and semi i ial solu ion (u, 0) exis ;
3
–I µ∈(0, µ1(λ)) he e exis s a leas a semicoexis ence s a e (u, )
and he componen has dead co e;
–I µ∈(µ1(λ), µ2(λ)) he e exis s a leas a semicoexis ence s a e;
–I µ∈(µ2(λ),∞) he e exis s a leas a semicoexis ence s a e
(u, ) and he componen uhas dead co e;
–I , mo eo e µ∗(λ)< µ∗(λ), hen i µ∈(µ∗(λ), µ∗(λ)) he e exis s
a leas a coexis ence s a e.
Analogous esul s can be ob ained when we ix he pa ame e µ. I ’s wo h
men ioning ha he exis ence o µ1(λ)>0 was shown in [14] when all
he coe icien s we e posi i e cons an s. To ou knowledge, he exis ence o
µ2(λ)>0 is new. In Rema k 3.1 we gi e a biological in e p e a ion o his
esul .
In Sec ions 4 and 5 we s udy he uniqueness o coexis ence s a es o (3).
Fo ha we use he ixed poin index. Obse e ha because m, n > 1 he
linea iza ion o (3) a ound he i ial o semi i ial solu ions do no exis , so
we canno apply he esul s in [11] (see also [25] and [28]) o compu e hei
indices. So, we will build app op ia e homo opies o ha . To compu e
he index o a coexis ence s a e we can use a linea iza ion. In his case
he linea iza ion o (3) a ound a coexis ence s a e leads us o a eigen alue
p oblem o he o m
(LU+MU =σU in Ω,
U= 0 on ∂Ω, (5)
whe e L= diag(L1, L2) and M= (mij), 1 ≤i, j ≤2 wi h mij ≥0 o
i6=jand mij blowing up nea ∂Ω in a con olled way. Following [16] and
[28] we de ine a speci ic o de and es ablish he exis ence o he p incipal
eigen alue o (5) as well as a cha ac e iza ion o i s posi i i y by means
he exis ence o a supe solu ion. Now, we p o e ha , again wi h ixed
λ > 0, he e exis s a unique coexis ence s a e when µbelongs o a subse o
(µ∗(λ), µ∗(λ)). Fu he mo e, i m=nand a, d a e s ic ly posi i e unc ions
we ha e uniqueness o coexis ence s a e i bMo cMis small. The esul s
abou uniqueness o coexis ence s a e o (3) a e also, we belie e, new.
2. P elimina ies. The degene a e logis ic equa ion
We conside he Banach space X:= C1
0(Ω) o de ed by i s cone o non-
nega i e unc ions P, whose in e io is
in (P) := {u∈X:u(x)>0 o all x∈Ω and ∂u/∂n < 0 on ∂Ω},
whe e ndeno es he ou wa d uni no mal on ∂Ω. We say ha u∈Xis
nonnega i e, u≥0, i u∈P, and uis posi i e, u > 0, i u∈in (P).
4
Gi en q∈L∞(Ω) and Lan ope a o o he o m (2), we deno e by σ1(L+q)
he p incipal eigen alue o L+qsubjec o homogeneous Di ichle bounda y
condi ions. Mo eo e , i we deno e by ϕ∈in P he unique posi i e eigen-
unc ion associa ed wi h σ1(L+q) no malized such ha kϕk∞= 1, hen i
is well known ha ∂ϕ
∂ν <0 on ∂Ω, (6)
o νany di ec ion ou o Ω. Recall ha as posi i e cons an s a e supe so-
lu ions o L, hen
σ1(L)>0.(7)
Finally, o ∈Y:= C0(Ω) we w i e
M:= max
x∈Ω
(x), L:= min
x∈Ω
(x).
2.1. Exis ence o solu ions In his sec ion we s udy he semi i ial
solu ions o (3). Obse e ha i he solu ions o (3) a e o he o m (u, 0)
and (0, ), hen sa is y equa ions o he ollowing ype
(Lw = (x)wq−g(x)wpin Ω,
w= 0 on ∂Ω,(8)
whe e Lis an ope a o o he o m (2), , g ∈C1(Ω) wi h g≥0, g6≡ 0,
can change sign and qand psa is y
(H) 0 < q < 1, p > q.
Ou i s esul gi es us he exis ence o nonnega i e solu ion o (8) and lis s
some use ul p ope ies. Fo a p oo o his esul see [15] o ins ance.
Theo em 2.1. Assume (H). The ollowing asse ions a e ue:
1. The e exis s a maximal nonnega i e and non i ial solu ion o (8) i ,
and only i , M>0. We deno e i by θ[L,q,p, ,g].
2. The ollowing es ima es hold:
θ[L,q,p, ,g](x)≤ 1/(1−q)
Meq/(1−q)
Me(x)x∈Ω,
(θ[L,q,p, ,g])M≤( MeM)1/(1−q),(9)
whe e e∈C2(Ω) is he unique solu ion o
(Le = 1 in Ω,
e= 0 on ∂Ω.(10)
3. I w∈C1(Ω) is a nonnega i e subsolu ion o (8), hen w≤θ[L,q,p, ,g].
5

4. Le i∈C1(Ω),i= 1,2be such ha 1≤ 2, hen θ[L,q,p, 1,g]≤
θ[L,q,p, 2,g].
5. I L>0, hen any nonnega i e solu ion o (8) is posi i e. Mo eo e ,
in his case he e exis s a unique posi i e solu ion and i sa is ies
εϕ(x)≤θ[L,q,p, ,g](x)x∈Ω,(11)
whe e εis he unique posi i e oo o
σ1(L)ε1−q+gMεp−q= L.(12)
Rema k. I we conside Las a eal pa ame e , hen i is easy o p o e
ha as L→ ∞,ε( L) = O( 1/(1−q)
L) when p≤1 and ε( L) = O( 1/(p−q)
L)
when p > 1.
2.2. Exis ence o dead co es In o de o s a e and p o e he main
esul , we need some p elimina y ones.
Lemma 2.2. Le R > 0and γ > 0. Conside he p oblem
(Lw =−Rwq−g(x)wpin Ω,
w=γon ∂Ω.(13)
Then, he e exis s a unique nonnega i e solu ion o (13).
P oo . Fo he exis ence we use he sub-supe solu ion me hod. Indeed,
i is easy o p o e ha (w, w) = (0, γ) is a sub-supe solu ion o (13). Fo
he uniqueness we can apply Theo em 2 in [1].
The ollowing echnical esul is undamen al in ou s udy. Mo eo e , i
gene alizes Lemma 7 in [30] and Lemma 2.5 in [3], whe e a simila esul
was p o ed when L=−∆ and g(x)≡0.
Lemma 2.3. We ix γ > 0and β > 2/(1 −q). Le δ0be such ha o all
x, x0∈IRNsuch ha 0≤ |x−x0| ≤ δ0
|x−x0|βq +β|x−x0|β−1L(|x−x0|)+
+β(1 −β)|x−x0|β−2
N
X
i,j=1
aij(x)Di(|x−x0|)Dj(|x−x0|)≥0.(14)
Then, o all 0< δ < dis (x0, ∂Ω), he unique nonnega i e solu ion, w, o
(13) in B(x0, δ)is such ha w(x0) = 0 p o ided ha
R≥µγ
min{δ, δ0}β¶1−q
.(15)
6
Rema k. Obse e ha since β > 2/(1−q), hen βq < β−2< β−1, and
so he exis ence o δ0sa is ying (14) is gua an eed. Mo eo e , since β > 2
(14) can be conside ed in a classical sense.
P oo . Conside he unc ion
Φ(x) := (Φ1(x) := R1/(1−q)|x−x0|βi x∈B(x0, δ0),
Φ2(x) := R1/(1−q)δβ
0i x∈B(x0, δ) B(x0, δ0),
wi h Φ ≡Φ1i δ≤δ0. By he choice o β, we ha e ha Φ1∈H2(B(x0, δ0)).
Mo eo e , ∂Φ1
∂nL
≥0,on ∂B(x0, δ0),
whe e nLs ands o he cono mal associa ed wi h L, i.e., (nL)i:= PN
j=1 aijnj.
Indeed, o x∈∂B(x0, δ0) we ha e
∂Φ1
∂nL
(x) = R1/(1−q)β|x−x0|β−3(
N
X
i,j=1
aij(x)(xi−xi
0)(xj−xj
0)) ≥0.
Mo eo e ,
L(Φ1) + RΦq
1+g(x)Φp
1=R1/(1−q)(β|x−x0|β−1L(|x−x0|)+
+β(1 −β)|x−x0|β−2
N
X
i,j=1
aijDi(|x−x0|)Dj(|x−x0|))+
+RRq/(1−q)|x−x0|βq +g(x)Rp/(1−q)|x−x0|βp ≥
≥R1/(1−q)(|x−x0|βq +β|x−x0|β−1L(|x−x0|)+
β(1 −β)|x−x0|β−2
N
X
i,j=1
aijDi(|x−x0|)Dj(|x−x0|)) ≥0,
by (14). In B(x0, δ) B(x0, δ0), we ha e ha
L(Φ2) + RΦq
2+g(x)Φp
2≥0.
Finally, in ∂B(x0, δ), Φ is bigge han γp o ided ha (15) holds.
Hence, we can apply Lemma I.1 in [4] and conclude ha Φ is a supe solu ion
o (13) in B(x0, δ). This comple es he p oo .
Fo R > 0, we de ine he se
N(R) := {x∈Ω : −(x)≥R}={x∈Ω : (x)≤ −R},
whe e ±(x) := max{± (x),0}. Assume ha ±6≡ 0. The main esul o
his sec ion is he ollowing one.
Theo em 2.4. Assume ha he e exis s R > 0such ha
7
1.
δR:= µ MeM
R¶1/(β(1−q))
≤δ0,
2.
M(R) := {x∈N(R) : dis (x, ∂N(R) ∂Ω) ≥δR} 6=∅.
Then, he e exis s a dead co e o any nonnega i e solu ion wo (8). Mo e-
o e , we ha e
M(R)⊂Ω0={x∈Ω : w(x) = 0}.
P oo . Le x0∈M(R), hen
B(x0, δR) := {x∈Ω : |x−x0|< δR} ⊂ N(R).(16)
We call z he unique nonnega i e solu ion o (13) in B(x0, δR) wi h γ=
( MeM)1/(1−q). Then, by (16) we ha e ha
Lθ[L,q,p, ,g]≤ −Rθq
[L,q,p, ,g]−g(x)θp
[L,q,p, ,g]in B(x0, δR),
which implies ha
L(z−θ[L,q,p, ,g])≥R(θq
[L,q,p, ,g]−zq) + g(x)(θp
[L,q,p, ,g]−zp) in B(x0, δR),
and by (9) and he choice o γwe ge
z≥θ[L,q,p, ,g]on ∂B(x0, δR).
Hence, i we deno e by Ω1:= {x∈ B(x0, δR) : z(x)< θ[L,q,p, ,g](x)} hen
L(z−θ[L,q,p, ,g])≥0 in Ω1,
z−θ[L,q,p, ,g]≥0 on ∂Ω1∩∂B(x0, δR),
z−θ[L,q,p, ,g]= 0 on ∂Ω1∩ B(x0, δR).
The maximum p inciple implies ha z≥θ[L,q,p, ,g]in B(x0, δR). Finally, we
can apply Lemma 2.3 because δRsa is ies (15). This inishes he p oo .
As consequence o he abo e esul , we ha e
Co olla y 2.5. Any nonnega i e solu ion o (8) has a dead co e p o-
ided ha Mis su icien ly small.
P oo . I is su icien o epea he p oo o Rema k 2.13 in [14] and o
ake accoun ha δR→0 as M→0.
8
3. Exis ence o nonnega i e solu ions
He ea e we w i e
θ[L1, ,g]:= θ[L1,1/m,2/m, ,g], θ[L2, ,g]:= θ[L2,1/n,2/n, ,g].
The ollowing esul gi es us a necessa y and su icien condi ion o ob ain
semicoexis ence s a es.
Theo em 3.1. P oblem (3) has a semicoexis ence s a e i , and only i ,
λ > 0and µ > 0.
P oo . By Theo em 2.1 3) i ollows ha
u≤θ[L1,λ,a], ≤θ[L2,µ,d].(17)
So, i λ≤0, again by Theo em 2.1 1) we ob ain ha u≡0. Analogously, i
µ≤0, ≡0.
Assume now ha λ > 0 and µ > 0. In his case, we ha e ha
A(x) := λ−b(x)θ1/n
[L2,µ,d](x)B(x) := µ−c(x)θ1/m
[L1,λ,a](x) (18)
sa is y AM=λ > 0 and BM=µ > 0. We conside he pai
(u, u) = (θ[L1,A,a], θ[L1,λ,a]),( , ) = (θ[L2,B,d], θ[L2,µ,d]).
By de ini ion o Aand Band Theo em 2.1 i ollows ha u≤u, ≤
and ha uand a e nonnega i e and non i ial unc ions. Finally, i is no
ha d o p o e ha he pai (u, u)−( , ) is a sub-supe solu ion o (3). This
comple es he p oo .
The ollowing esul p o ides us wi h condi ions which assu e he exis-
ence o coexis ence s a es as well as bounds o hem.
Theo em 3.2. I λand µsa is y
λ > (b(x)θ1/n
[L2,µ,d])M, µ > (c(x)θ1/m
[L1,λ,a])M,(19)
hen, (3) possesses a coexis ence s a e. Mo eo e , o any coexis ence s a e
(u, )o (3) we ha e he ollowing es ima es: i λ > (b(x)θ1/n
[L2,µ,d])M hen
ε1ϕ1≤θ[L1,A,a]≤u≤θ[L1,λ,a]≤λm/(m−1)(e1)1/(m−1)
Me1,(20)
and i µ > (c(x)θ1/m
[L1,λ,a])M, hen
ε2ϕ2≤θ[L2,B,d]≤ ≤θ[L2,µ,d]≤µn/(n−1)(e2)1/(n−1)
Me2,(21)
9
The ollowing esul will be used o compa e p incipal eigen alues o
di e en ma ices.
Lemma 4.6. Le A(x)=(aij(x)) and B(x)=(bij(x)) be wo ma ices
wi h aij, bij sa is ying (HM), bii ≥aii and aij ≥bji o i6=jwi h some
inequali y s ic . Then, σ1(L+A)< σ1(L+B).
P oo . Le ΦAÂ0 be he eigen unc ion associa ed wi h L+A. Then,
i is easy o show ha
(L+B−σ1(L+A)I)ΦAÂ0,
and so, ΦAis a s ic supe solu ion o L+B−σ1(L+A)I. Hence, by
Theo em 4.2 we deduce ha σ1(L+B−σ1(L+A)I)>0, whence he
conclusion ollows.
5. Uniqueness esul
Along his sec ion we assume ha λand µsa is y (19), and so he alidi y
o he s ong maximum p inciple is gua an eed. Indeed, by (21) we ge
u1/m(λ−b(x) 1/n)−a(x)u2/m ≥u1/m(λ−b(x)θ1/n
[L2,µ,d])−a(x)u2/m,
and so, by (19), he e exis s a posi i e cons an Msuch ha
u1/m(λ−b(x) 1/n)−a(x)u2/m +Mu ≥0,(32)
whence i ollows ha i (u, ) is a non-nega i e solu ion o (3) wi h u6≡,
hen u(x)>0 o all x∈Ω. Simila ly we can eason wi h he second
equa ion in (3).
In his sec ion we ob ain a uniqueness esul o a coexis ence s a e o
(3). In o de o ge he esul we use he ixed poin index in cones.
Fixed M > 0 ob ained in (32), conside he ope a o K:X27→ X2
de ined by
K(u, ) := Ã(L1+M)−1(u1/m(λ−a(x)u1/m −b(x) 1/n) + Mu)
(L2+M)−1( 1/n(µ−d(x) 1/n −c(x)u1/m) + M )!,
whe e (Li+M)−1,i= 1,2, s ands o he in e se on he ope a o Li+M
in Ω unde homogeneous Di ichle bounda y condi ions. Obse e ha by
(7), σ1(Li+M)>0 and so (Li+M)−1is well-de ined and i is a compac
ope a o . Thanks o he choice o M, see (32), Kis a posi i e ope a o
whose ixed poin s a e componen wise nonnega i e solu ions o (3).
16

On he o he hand, by (20) and (21), he e exis Ri>0, i= 1,2, such ha
o e e y (u, ) coexis ence s a es o (3)
kuk∞≤R1:= (λ(e1)M)m/(m−1),k k∞≤R2:= (µ(e2)M)n/(n−1).
So, he ixed poin index o Ko e Bwi h espec o he cone P×Pis well
de ined, whe e
B:= {(u, )∈P2:kuk∞≤R1+ 1,k k∞≤R2+ 1}.
Now, we a e going o compu e his index in some cases.
P oposi ion 5.1. Assume ha λand µsa is y (19). The ollowing
asse ions a e ue:
1. iP×P(K,B) = 1;
2. iP×P(K,(0,0)) = 0;
3. iP×P(K,(θ[L1,λ,a],0)) = iP×P(K,(0, θ[L2,µ,d])) = 0.
P oo . 1.) Fi s ly, we de ine G1:X7→ Xby
G1(u) := (L1+M)−1(u1/m(λ−a(x)u1/m) + Mu)
By (9), aking Bu:= {u∈P:kuk∞≤R1+ 1} he ixed poin index o G1
o e Buis well-de ined. Applying Lemma 12.1 in [2] i can be p o ed ha
iP(G1, Bu) = 1.(33)
Indeed, i he e exis ≥1 and u∈Psuch ha kuk∞=R1+ 1 and
G1(u) = u, hen
L1u≤u1/m(λ
−a(x)
u1/m),
and so,
kuk∞≤µλ
¶m/(m−1)
(e1)m/(m−1)
M≤R1< R1+ 1.
Analogously,
iP(G2, B ) = 1 (34)
wi h G2( ) := (L2+M)−1( 1/n(µ−d(x) 1/n) + M ) and B := { ∈P:
k k∞≤R2+ 1}.
Conside he ope a o H1: [0,1] ×X27→ X2de ined by
H1( , u, ) := Ã(L1+M)−1(u1/m(λ−a(x)u1/m − b(x) 1/n) + Mu)
(L2+M)−1( 1/n(µ−d(x) 1/n − c(x)u1/m) + M )!.
17
Obse e ha by (20) and (21) any ixed poin o H1belongs o B. So, i
ollows by homo opy in a iance, (33) and (34) ha
iP×P(K,B) = iP×P(H1(1,·),B) = iP×P(H1(0,·),B)
=iP(G1, Bu)·iP(G2, B ) = 1.
We now p o e 2). Le ψi∈Y,i= 1,2, be such ha ψi>0 in Ω. We de ine
H2( , u, ) := Ã(L1+M)−1(u1/m(λ−a(x)u1/m −b(x) 1/n) + Mu + ψ1)
(L2+M)−1( 1/n(µ−d(x) 1/n −c(x)u1/m) + M + ψ2)!.
We claim ha he e exis s δ > 0 such ha
(u, )6=H2( , u, ),∀ ∈[0,1],∀(u, )∈ Nδ,(35)
whe e Nδ:= {(u, )∈P2:kuk∞≤δ, k k∞≤δ} {(0,0)}. Assume he e
exis sequences (u , ) o unc ions and ∈[0,1] such ha (u , )→(0,0)
as → ∞ and
(u , ) = H2( , u , ).
Since λ > 0 and k k∞→0, he e exis s 0∈IN such ha (λ−b(x) 1/n
)L>
0 o ≥ 0. So, he s ong maximum p inciple is sa is ied in he i s
equa ion, and so u >0. Le K > 0 be such ha K≥σ1(L1). Since
ku k∞→0, he e exis s 1∈IN such ha o ≥ 1we ha e
L1u =u1/m
(λ−b(x) 1/n
)−a(x)u2/m
+ ψ1> Ku ,
and hence σ1(L1−K)>0, a con adic ion.
Thus, by (35) he homo opy is admissible and we ge
iP×P(K,(0,0)) = iP×P(K,Nδ) = iP×P(H2(0,·),Nδ)
=iP×P(H2(1,·),Nδ) = 0,
his las equali y ollows by (35).
I emains o p o e 3). Le ψ∈Ybe such ha ψ > 0 in Ω. We de ine
ano he ope a o
H3( , u, ) := Ã(L1+M)−1(u1/m(λ−a(x)u1/m −b(x) 1/n) + Mu)
(L2+M)−1( 1/n(µ−d(x) 1/n −c(x)u1/m) + M + ψ)!.
We claim ha he e exis s δ > 0 such ha
(u, )6=H3( , u, ),∀ ∈[0,1],∀(u, )∈ Mδ,(36)
whe e Mδ:= {(u, )∈P2:ku−θ[L1,λ,a]k∞≤δ, k k∞≤δ} {(θ[L1,λ,a],0)}.
Assume he e exis sequences (u , )→(θ[L1,λ,a],0) as → ∞ and ∈[0,1]
such ha
(u , ) = H3( , u , ).
18
Since u ≤θ[L1,λ,a], and µ > (c(x)θ1/m
[L1,λ,a])Mi ollows ha
µ−c(x)u1/m
(x)≥µ−c(x)θ1/m
[L1,λ,a]>0.(37)
Le K > 0 be such ha K≥σ1(L2). Then, by (37) he e exis s 0∈IN
such ha o ≥ 0we ha e
L2 = 1/n
(µ−c(x)u1/m
)−d(x) 2/n
+ ψ > K ,in Ω,
and hence σ1(L2−K)>0, a con adic ion.
Thus, by (36) he homo opy is admissible and we ge
iP×P(K,(θ[L1,λ,a],0)) = iP×P(K,Mδ) = iP×P(H3(0,·),Mδ)
=iP×P(H3(1,·),Mδ) = 0.
Analogously, i can be ea ed he solu ion (0, θ[L2,µ,d]).
Now, le (u0, 0) be a coexis ence s a e o (3). We conside he ma ix
M(u0, 0):= (mij), i, j = 1,2, which is ela ed o he linea iza ion o (3)
abou (u0, 0), whe e
m11 =−1
mu1/m−1
0(λ−2a(x)u1/m
0−b(x) 1/n
0),
m12 =1
nb(x)u1/m
0 1/n−1
0,
m21 =1
mc(x) 1/n
0u1/m−1
0,
m22 =−1
n 1/n−1
0(µ−2d(x) 1/n
0−c(x)u1/m
0).
(38)
Obse e ha since (u0, 0) is a coexis ence s a e, by (20) and (21) he e
exis s k0>0 such ha
k0dis (x, ∂Ω) ≤u0, k0dis (x, ∂Ω) ≤ 0,
hen M(u0, 0)sa is ies (HM), so ha σ1(L+M(u0, 0)) makes sense.
The gene al uniqueness esul eads
Theo em 5.2. Assume ha λand µsa is y (19) and σ1(L+M(u0, 0))>
0 o any (u0, 0)coexis ence s a e o (3). Then, (3) possesses a unique
coexis ence s a e.
P oo . Recall ha by P oposi ion 3.3, i λand µsa is y (19) hen any
nonnega i e solu ion o (3) is a coexis ence s a e. We claim ha i (u0, 0)
is a coexis ence s a e o (3), hen
iP×P(K,(u0, 0)) = 1.(39)
19
Assume ha we ha e p o ed (39), hen since Kis a compac ope a o , i
possesses a ini e numbe o coexis ence s a es, say (ui, i), i= 1, . . . , .
Then,
iP×P(K,B) = iP×P(K,(0,0)) + iP×P(K,(θ[L1,λ,a],0))
+iP×P(K,(0, θ[L2,µ,d])) +
X
i=1
iP×P(K,(ui, i))
and so, by P oposi ion 5.1 and (39),
1 = 0 + 0 + 0 + ,
whence he conclusion now easily ollows.
I emains o p o e (39). Le h∈C1(Ω) be such ha h e i ies ha
|h(x)|dis (x, ∂Ω)2−α≤K o some α∈(0,2], K > 0 and
h≥max{0, m11, m22},(40)
whe e m11 and m22 a e de ined in (38). We de ine he ope a o
T(u, ) := Ã(L1+h)−1(u1/m(λ−a(x)u1/m −b(x) 1/n) + hu)
(L2+h)−1( 1/n(µ−d(x) 1/n −c(x)u1/m) + h )!.
Obse e ha (Li+h)−1exis s because h≥0 and so σ1(Li+h)>0.
By he Le ay-Schaude o mula, iP×P(T,(u0, 0)) = (−1)ξ, whe e ξis he
sum o he mul iplici ies o he eigen alues o D(u, )T(u0, 0) la ge han
one, being D(u, )T(u0, 0) he linea iza ion o Tabou (u0, 0). I is clea
ha
D(u, )T(u0, 0) = diag((L1+h)−1,(L2+h)−1)(−M(u0, 0)+ diag(h, h)),
whe e M(u0, 0)is de ined by (38). I is no ha d o p o e ha i > 1 is an
eigen alue o D(u, )T(u0, 0), hen
σ1(L+M(u0, 0)+B) = 0,(41)
whe e
B=Ã(m11 −h)(1
−1) m12(1
−1)
m21(1
−1) (m22 −h)(1
−1) !
Since > 1, by (40) and Lemma 4.6 we ge
σ1(L+M(u0, 0)+B)> σ1(L+M(u0, 0))>0,
con adic ing (41).
The ollowing esul p o ides us wi h a su icien condi ion o σ1(L+
M(u0, 0))>0 o be hold.
20
P oposi ion 5.3. Assume ha m=n,a(x), d(x)>0 o all x∈Ω,λ
and µsa is y (19) and ha o any (u0, 0)coexis ence s a e o (3)
µb
a¶Mµc
d¶Mµu0
0¶(2−m)/m
Mµ 0
u0¶(2−m)/m
M
<1.(42)
Then, (3) possesses a unique coexis ence s a e.
P oo . Le
Φ := (αu1/m
0,−β 1/m
0)∈(C2(Ω) ∩C0
0(Ω))2,
wi h α, β > 0 o be chosen. We will show ha Φ is a supe solu ion in he
sense o De ini ion 4.1 o L+M(u0, 0)i (42) holds. P oposi ion 4.3 and
Theo em 5.2 will comple e he p oo .
Fi s ly, obse e ha Φ Â0. In o de o show ha Φ is a supe solu ion
o L+M(u0, 0)we ha e o p o e ha ( o he i s equa ion)
L1(αu1/m
0) + m11(x)αu1/m
0+m12(x)(−β 1/m
0)>0,(43)
whe e m11 and m12 a e de ined in (38). Taking in o accoun he ac ha
L1(u1/m
0) = 1
mu1/m−1
0[(1 −1
m)u−1
0
N
X
i,j=1
a1
ijDiu0Dju0+L1u0],
o p o e (43) i su ices ha
a(x)u2/m−1
0> b(x) 2/m−1
0·β
α, o all x∈Ω.
Analogously, o he second equa ion i is su icien ha
d(x) 2/m−1
0> c(x)u2/m−1
0·α
β, o all x∈Ω.
Now, by (42) i is easy o show ha he e exis αand βsa is ying he abo e
inequali ies.
The ollowing esul p o ides us ano he su icien condi ion o ob ain a
uniqueness esul .
P oposi ion 5.4. Assume ha λand µsa is y (19) and ha o any
coexis ence s a e (u0, 0)o (3) he ollowing inequali ies hold o all x∈Ω,
λ(1 −1
m) + a(x)u1/m
0(x)( 2
m−1) > b(x)µ1 + 1
n−1
m¶ 1/n
0(x),
µ(1 −1
n) + d(x) 1/n
0(x)( 2
n−1) > c(x)µ1 + 1
m−1
n¶u1/m
0(x).
(44)
Then, (3) possesses a unique coexis ence s a e.
21

P oo . Taking
Φ := (u0,− 0),
i su ices o p o e ha Φ Â0 is a supe solu ion o L+M(u0, 0)p o ided
ha (44) and apply again P oposi ion 4.3 and Theo em 5.2. Fo he second
equa ion, Φ is a supe solu ion i
L2(− 0) + m21(x)(u0) + m22(x)(− 0)<0,
whe e m21 and m22 a e de ined in (38). Fo obse e ha
L2(− 0) + m21(x)(u0) + m22(x)(− 0)
= 1/n
0³µ³1
n−1´+d(x) 1/n
0³1−2
n´+c(x)u1/m
0³1 + 1
m−1
n´´<0,
p o ided ha (44) holds. Simila ly we can eason wi h he i s equa ion.
Now, we will use he uppe es ima es o (20) and (21) gi ing su icien
condi ions o he uniqueness o coexis ence s a e in e ms o se e al coe i-
cien s in ol ed in he model se ing.
Co olla y 5.5. Assume ha m=n,a(x), d(x)>0 o x∈Ω,λand
µsa is y (19) and
Ã(e1)1/(m−1)
M
ε1
(e2)1/(m−1)
M
ε2!
2−m
mµe1
ϕ2¶
2−m
m
Mµe2
ϕ1¶
2−m
m
M
(λµ)
2−m
m−1<aLdL
bMcM
,
(45)
whe e ε1and ε2a e de ined in (22). Then, (3) possesses a unique coexis ence
s a e.
P oo . By (20) and (21) we ha e ha
µu0
0¶M
≤λm/(m−1)(e1)1/(m−1)
M
ε2µe1
ϕ2¶M
,
and
µ 0
u0¶M
≤µm/(m−1)(e2)1/(m−1)
M
ε1µe2
ϕ1¶M
,
and so, (42) is sa is ied i (45) holds. I su ices o apply P oposi ion 5.3.
Co olla y 5.6. Assume ha some o he ollowing se s o inequali y,
1 o 4, holds:
1. I 1< m, n ≤2,
bMµ1 + 1
n−1
m¶µ1/(n−1)(e2)1/(n−1)
M< λ(1 −1
m),
cMµ1 + 1
m−1
n¶λ1/(m−1)(e1)1/(m−1)
M< µ(1 −1
n),
22
2. I 1< n ≤2and m > 2,
bM³1 + 1
n−1
m´µ1/(n−1)(e2)1/(n−1)
M+ (1 −2
m)aMλ1/(m−1)(e1)1/(m−1)
M
< λ(1 −1
m),
cM³1 + 1
m−1
n´λ1/(m−1)(e1)1/(m−1)
M< µ(1 −1
n),
3. I 1< m ≤2and n > 2,
cM³1 + 1
m−1
n´λ1/(m−1)(e1)1/(m−1)
M+ (1 −2
n)dMµ1/(n−1)(e2)1/(n−1)
M
< µ(1 −1
n),
bM³1 + 1
n−1
m´µ1/(n−1)(e2)1/(n−1)
M< λ(1 −1
m),
4. I m > 2and n > 2,
bM³1 + 1
n−1
m´µ1/(n−1)(e2)1/(n−1)
M+ (1 −2
m)aMλ1/(m−1)(e1)1/(m−1)
M
< λ(1 −1
m),
cM³1 + 1
m−1
n´λ1/(m−1)(e1)1/(m−1)
M+ (1 −2
n)dMµ1/(n−1)(e2)1/(n−1)
M
< µ(1 −1
n),
hen, (3) possesses a unique coexis ence s a e.
P oo . Reasoning as in he p oo o Co olla y 5.5, i is su icien o
apply (20), (21) and P oposi ion 5.4.
Rema k. 1. Obse e ha when m= 1, (42) is he condi ion ob ained
in Theo em 4.2 in [28] and Theo em 4.8 in [22]. Mo eo e , when
m=n, and aand da e posi i e, we ob ain uniqueness p o ided ha
bMo cMis small.
2. The (λ, µ)- egions de ined in Co olla y 5.6 a e subse s o he coex-
is ence egion ob ained in Theo em 3.2. Simila condi ions o hose
imposed in Figu e 1 assu e he exis ence o hese sub egions.
6. Conclusions
We ha e s udied he se o non-nega i e solu ions o a spa ially he e o-
geneous Lo ka-Vol e a compe i ion model wi h degene a e di usion. Basi-
cally, we ha e ound h ee di e ences wi h he espec o he non-degene a e
(linea ) case:
1. In he degene a e case all he non-nega i e solu ions a e bounded,
unlike he linea case in which a-p io i bounds a e los o some alues
o he da a o he p oblem.
23
2. In he degene a e case a new kind o non-nega i e solu ions appea s:
non-nega i e and non i ial solu ions ha anish in a egion o he
habi a o he species. We ob ain su icien condi ions in e ms o
some pa ame e s in ol ed in he se ing o he model ensu ing he
exis ence o non-exis ence o such kind o solu ions.
3. Unlike he non-degene a e case, in ou model when he compe i ion
be ween he species is “s ong” nei he species d i es o he o he o
ex inc ion.
Finally, we ha e ob ained uniqueness o posi i e solu ion o he p oblem
unde some condi ions on he da a o he p oblem.
Acknowledgmen s: The au ho is in g a e ul o P o esso M. Delgado
o his help ul commen s. He hanks o MCYT o Spain o esea ch sup-
po unde g an BFM2000-0797. Finally, he would like o acknowledge he
anonymous e e ee o use ul ema ks which imp o ed his pape .
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