ON THE EXISTENCE OF DEAD CORES
FOR DEGENERATE LOTKA-VOLTERRA MODELS
Manuel DELGADO and An onio SU´
AREZ
Dp o. de Ecuaciones Di e enciales y An´alisis Num´e ico, Facul ad de Ma em´a icas,
C/ Ta ia s/n, Uni e sidad de Se illa, 41012-Se illa, Spain.
e-mail: delgado@nume .us.es and sua ez@nume .us.es
Abs ac
In his wo k we s udy he exis ence, uniqueness and quali a i e p ope ies o nonnega i e
solu ions o he Lo ka-Vol e a models wi h nonlinea di usion unde homogeneous Di ichle
bounda y condi ions. We conside he h ee ypical in e ac ions: p ey-p eda o , compe i ion
and symbiosis. Unlike he linea di usion models, non i ial nonnega i e solu ions can exis
which a e no s ic ly posi i e. Su icien condi ions in e ms o he coe icien s in ol ed in
he se ing o he models a e gi en assu ing ha one species (o bo h) does no su i e on
a se o i s habi a (called “dead co e”) o posi i e measu e.
AMS Classi ica ion Codes: 35B50, 35B99, 35J55, 35K57, 92D25.
Keywo ds and ph ases: degene a e Lo ka-Vol e a models, coexis ence s a es, dead co e se .
1
1 In oduc ion
In his pape we a e in e es ed in nonnega i e solu ions o he ollowing sys em





−d1∆Um=U(A−BU ±CV ) in Ω,
−d2∆Vn=V(D−EV ±FU) in Ω,
U=V= 0 on ∂Ω,
(1)
whe e Ω is a bounded domain o IRN,N≥1, wi h a smoo h bounda y ∂Ω and m, n, d1, d2, B,
C,E,Fa e posi i e cons an s wi h m, n ≥1 and A, D ∈IR. The p oblem (1) models he in-
e ac ions be ween wo species, wi h popula ion densi ies U(x) and V(x), inhabi ing he egion
Ω. Mo eo e , 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. Nonlinea di -
usion a ises mainly in he po ous media equa ion and i was in oduced in popula ion dynamics
in [16]. We e e o [12] and he e e ences he ein o de ails abou he model. Fo he p esen
model Uand Vcan in e ac in h ee di e en ways indica ed by he sign o he las e ms in
he equa ions: i bo h o hem a e nega i e hen Uand Vcompe e; i hey a e posi i e hen U
and Vco-ope a e; and i , o example, he sign is posi i e in he i s equa ion and nega i e in
he second one hen Uand V ep esen he p eda o and p ey popula ions espec i ely.
To s udy (1), we make an app opia e change o a iables (see [12]) and ob ain





−∆wm=w(λ−w±bz) in Ω,
−∆zn=z(µ−z±cw) in Ω,
w=z= 0 on ∂Ω,
(2)
whe e λ, µ ∈IR and b, c > 0.
When m=n= 1, (2) is he classical Lo ka-Vol e a model which has been widely s udied
in he las yea s: see [4], [6], [8], [9], [15], [23], [26], [28] in compe i ion, [4], [9], [10], [21], [25] in
p eda o -p ey and [11], [19], [22], [24], [27] in symbiosis, o ins ance.
When m, n > 1 he e exis s an impo an change in he beha iou o he solu ions o (2).
Mo e p ecisely, i is possible ha one species (o bo h) is nonnega i e bu no s ic ly posi i e,
i.e. he e exis some subse s o Ω wi h s ic ly posi i e measu es whe e he species does no
su i e. We call hem “dead co es” (see [14]). Fo ha , we dis inguish wo ypes o non i ial
nonnega i e solu ions o (2); hose wi h bo h componen s posi i e, he coexis ence s a es, and
hose whe e a leas one componen could ha e a dead co e, he semi-coexis ence s a es.
We now desc ibe he dis ibu ion and he con en s o his wo k. In Sec ion 2 we s udy he
single bounda y alue p oblem
(−∆wm=w(a(x)−dw) in Ω,
w= 0 on ∂Ω, (3)
which appea s when one o he species is ze o and whe e a∈Cα(Ω), α∈(0,1), m≥1 and
d≥0. I m= 1 i is known ha (3) admi s a unique posi i e solu ion i and only i σΩ
1(−a)<0,
whe e σΩ
1(q) s ands o he p incipal eigen alue o he p oblem
(−∆w+q(x)w=σw in Ω,
w= 0 on ∂Ω,
2
wi h q∈L∞(Ω). The pa icula case m > 1 and d= 0 has been s udied in [1], [2], [3] and [17].
Thus, we ocus on he case m > 1 and d > 0 which has been p e iously analyzed in [17] and
[30]. To s a e ou main esul s, we need he ollowing no a ion. Fo any a∈Cα(Ω) we deno e
aL:= min
x∈Ω
a(x)aM:= max
x∈Ω
a(x).
In Theo em 1 we ob ain some esul s conce ning exis ence and uniqueness o nonnega i e solu-
ions o (3) which can be summa ized as ollows:
•I aL≥0 and 1 < m ≤2 hen he e exis s a unique posi i e solu ion o (3).
•I aM>0 hen he e exis s a leas a nonnega i e solu ion o (3).
•I aM≤0 hen (3) does no admi a nonnega i e solu ion.
Pa icula a en ion is paid o he exis ence o a dead co e o solu ions o (3). We show:
•I ei he aL>0 o aL= 0 and 1 < m ≤2, any solu ion o (3) is posi i e and so i has no
a dead co e.
•I aL= 0, m > 2 and A0:= in {x∈Ω : a(x)=0} 6=∅, he e exis s a posi i e cons an
C=C(N, d, ε) such ha i aM< C hen any nonnega i e solu ion o (3) has a dead co e,
whe e ε > 0 is he adius o a ball con ained in A0.
When achanges sign, we w i e a (x) := a+(x)− a−(x) whe e a±(x) := max{0,±a(x)}and o
any > 0 and R > 0 we conside he se s
N(R, ) := {x∈Ω : a−(x)≥R
}, M(R, ) := {x∈N(R, ) : dis (x, ∂N(R, ) ∂Ω) ≥CR},
wi h
CR:= sN
2Z(a+
M
d)m
0
ds
[Zs
0
(Rν1/m +dν2/m)dν]1/2.
Wi h his no a ion, we ob ain ha i M(R, )6=∅ o some > 0 and R > 0 hen any nonnega i e
solu ion o (3) has a dead co e. Mo eo e , we conside as pa ame e and we ob ain in o ma ion
abou he quali a i e beha iou o he nonnega i e solu ions o (3). Ou esul s in his di ec ion
can be s a ed in he ollowing gene al e ms:
•The e exis s 0>0 such ha i ≥ 0 hen any nonnega i e solu ion o (3) has a dead
co e.
•I 1 < m ≤2 he e exis s 1>0 such ha i 0 < ≤ 1 hen any nonnega i e solu ion o
(3) is posi i e.
•I m > 2 he e exis d0>0 and 0(d0)>0 such ha i 0 < ≤ 0(d0) hen any nonnega i e
solu ion o (3) has a dead co e.
3
These esul s imp o e and ex end hose in [30] (See Rema k 5 and Sec ion 2).
In Sec ion 3 we analyze he sys em (2) in he h ee classical in e ac ions: p eda o -p ey, com-
pe i ion and symbiosis. The p ey-p eda o case has been in es iga ed in [7], [18], [20], [29], [31];
compe i ion one in [7], [29]; and las ly, he symbiosis one in [7]. We conside λand µas p incipal
pa ame e s and we s udy he se o alues o (λ, µ) o which (2) admi s a semi-coexis ence o
coexis ence s a e. On he o he hand, we ob ain a p io i bounds o nonnega i e solu ions o (2)
which a e he key o ge he non-exis ence egions o nonnega i e solu ions in he (λ, µ)-plane.
We should no ice ha ou esul s imp o e in some sense hose in he abo e men ioned wo ks
(see Rema ks 8, 9 and 11).
Now, we a e in e es ed in he exis ence o a dead co e o any nonnega i e solu ion o (2). This
has been p e iously s udied in he cases o p ey-p eda o and compe i ion in [18] and [29]. To ge
esul s conce ning his subjec , we apply he esul s o Sec ion 2 and ob ain o he p ey-p eda o
case:
•Fo ixed λ > 0, he e exis s µ0(λ)>0 such ha i 0 < µ < µ0(λ) any nonnega i e solu ion
o (2) has a dead co e.
•Fo ixed µ > 0, he e exis λ0(µ)<0 and ε > 0 such ha i λ≤λ0(µ) he e is no
nonnega i e solu ion o (2) and i λ∈(λ0(µ), λ0(µ) + ε) any nonnega i e solu ion o (2)
has a dead co e.
•Fo ixed λ, µ, b > 0, he e exis s c0>0 such ha i c > c0any nonnega i e solu ion o (2)
has a dead co e.
In he case o compe i ion, we ha e:
•Fo ixed λ > 0 ( esp. µ > 0), he e exis s µ0(λ)>0 ( esp. λ0(µ)>0) such ha i
0< µ < µ0(λ) ( esp. 0 < λ < λ0(µ)) any nonnega i e solu ion o (2) has a dead co e.
•Fo ixed λ, µ > 0 and c > 0 ( esp. b > 0), he e exis s b0>0 ( esp. c0>0) such ha i
b > b0( esp. c > c0) any nonnega i e solu ion o (2) has a dead co e.
Mo eo e , in his Sec ion we will gi e a biological in e p e a ion o hese esul s.
2 The logis ic equa ion wi h nonlinea di usion
In his pape we use he ollowing no a ion: Ω is a bounded domain in IRNwi h a smoo h enough
bounda y ∂Ω. Fo ixed α > 0, we conside he spaces U:= {w∈C2,α(Ω) : w= 0 on ∂Ω}
and V:= Cα(Ω) o de ed by hei cones o nonnega i e unc ions PU:= {w∈ U :w≥0}and
PV:= {w∈ V :w≥0}. We will w i e ≥gi −g∈P, > g i −g∈P−{0}and Àg
i −g∈˙
P, whe e ˙
Pdeno es he in e io o P. Mo eo e , o any ∈ V we deno e
M:= max
x∈Ω
(x) L:= min
x∈Ω
(x).
4
Le a∈ V,m > 1 and d > 0, we conside he logis ic equa ion wi h nonlinea di usion, namely
(−∆wm=a(x)w−dw2in Ω,
w= 0 on ∂Ω. (4)
Gi en q∈L∞(Ω), σΩ
1(q) s ands o he i s eigen alue o
(−∆w+q(x)w=σw in Ω,
w= 0 on ∂Ω, (5)
whose co esponding associa ed eigen unc ion ϕΩ
1[q] can be chosen such ha ϕΩ
1[q]À0 and
kϕΩ
1[q]k∞= 1. Mo eo e , due o he s ong maximun p inciple, we ha e
∂ϕΩ
1[q]
∂n <0 (6)
whe e nis he ou wa d uni no mal a ∂Ω. Finally, we w i e σΩ
1:= σΩ
1(0) and ϕΩ
1:= ϕΩ
1[0].
To s udy (4), we pe o m he change o a iables wm=uand we ob ain
(−∆u=a(x)u1/m −du2/m in Ω,
u= 0 on ∂Ω. (7)
Fo he exis ence and uniqueness o nonnega i e solu ions o (7), he main esul is he ollowing
one:
Theo em 1 Le a∈ V,d > 0and m > 1. The ollowing asse ions a e ue:
1. I aÀ0, hen he e exis s a solu ion u∈ U o (7) wi h uÀ0. Mo eo e , i m≤2 he
solu ion is unique.
2. I a > 0and m≤2, hen he e exis s a unique solu ion u∈ U o (7) wi h uÀ0.
3. I he e exis s x0∈Ωsuch ha a(x0)>0, hen he e exis s a leas a solu ion u∈ U o
(7) wi h u > 0.
Rema k 1 I a≤0 hen u= 0 is he unique nonnega i e solu ion o (7) owing o he maximum
p inciple. On he o he hand, obse e ha he unc ion acan change o sign in case 3.
P oo . We will apply he sub-supe solu ion me hod. By he egula i y o a he e exis x0∈Ω
and > 0 such ha
a(x)≥a0>0, o x∈B(x0, ),
whe e B(x0, ) is he ball o adius > 0 cen e ed a x0. Now, we de ine he unc ion
φ(x) := (ϕΩ0
1(x) i x∈Ω0,
0 i x∈Ω Ω0,
5

wi h Ω0= Ω in he case 1 and Ω0=B(x0, ) in he o he cases. We will show ha u=ρφ is a
subsolu ion o (7) wi h ρa posi i e cons an . Indeed, we can ake ρ > 0 such ha
−∆(ρϕΩ0
1)≤a(x)(ρϕΩ0
1)1/m −d(ρϕΩ0
1)2/m in Ω0.
By (6) we can use Lemma I.1 o [3] and conclude ha uis a subsolu ion o (7). As a supe solu ion
we pick u=M, wi h M > 0 a su icien ly la ge cons an . Mo eo e , i is easy o see ha
kuk∞≤M:= µaM
d¶m
(8)
o any solu ion uo (7). So, he e exis s a leas a solu ion u∈ U such ha u≤u≤u.
Clea ly uÀ0 in he i s case. Now, we show ha uÀ0 in he case 2 by applying he s ong
maximum p inciple. Le u > 0 be a solu ion o (7); i su ices o ind a cons an K > 0 such
ha (−∆ + K)u > 0, o equi alen ly
K > du2/m−1−a(x)u1/m−1,wi h u∈(0, M],(9)
which ollows om hypo heses and so uÀ0.
Fo he uniqueness we use [5]. We de ine he map
7→ g(x, ) := a(x) 1/m −d 2/m
=a(x) 1/m−1−d 2/m−1.
I is no ha d o p o e ha his applica ion is dec easing in he cases aÀ0 and m≤2 as well
as a > 0 and m < 2. I emains o p o e he uniqueness in he case m= 2 and a > 0. Le
u1and u2be wo a bi a y solu ions o (7). Now, he unc ion gis noninc easing, so by he
Rema k 1 o [5], i ollows ha he e exis s a cons an Csuch ha u1=Cu2. Hence,
a(x)u1/2
1−du1=−∆u1=−∆(Cu2) = C(a(x)u1/2
2−du2) = Ca(x)u1/2
2−du1
om which
a(x)C1/2u1/2
2(1 −C1/2) = 0,
and he e o e C= 1. This comple es he p oo .
•
Rema k 2 1.- When aL>0 he e exis s a unique posi i e solu ion o (7) p o ided ha
m > 2and aL
aM
>m−2
m−1.
Indeed, i can be p o ed ha he unc ion 7→ g(x, )de ined in he p oo o Theo em 1 is
dec easing.
2.- In he pa icula case a(x)≡λ=c e i is known, c . [13], ha he e exis s a unique posi i e
solu ion o (7) i and only i λ > 0.
3.- When aÀ0and m > 1, any solu ion o (7) is posi i e in Ω. In his case, i is easy o p o e
he exis ence o he cons an Ksa is ying (9).
6
¿F om he p io i bound (8) ollows he exis ence o a non i ial maximal solu ion o (7) which
we deno e by θ[a,d,m]. We will ake θ[a,d,m]= 0 i a≤0.
The nex esul will be e y use ul o compa e posi i e solu ions o di e en logis ic bounda y
alue p oblems.
P oposi ion 1 Le a, b ∈ V.
1. I uis a subsolu ion o (7), hen
u≤θ[a,d,m].
2. I a≤b, hen
θ[a,d,m]≤θ[b,d,m].
P oo . Le ube a subsolu ion o (7). As su icien ly la ge cons an s, say K > 0, a e supe solu-
ions, hen he e exis s a solu ion uo (7) such ha u≤u≤K. F om he maximali y o θ[a,d,m]
i ollows ha u≤θ[a,d,m]. Fo he second pa i is enough o p o e ha θ[a,d,m]is a subsolu ion
o (−∆w=b(x)w1/m −dw2/m in Ω,
w= 0 on ∂Ω,
and apply he p e ious esul .
•
2.1 Exis ence o a “dead co e”
In his subsec ion we shall conside he exis ence o a “dead co e” o solu ions uo (7), i.e., we
will show ha he se Ω0:= {x∈Ω : u(x) = 0}has a s ic ly posi i e measu e unde sui able
easily checked hypo heses on a,d,m, Ω and N.
Assume ha a > 0 and
A0:= in {x∈Ω : a(x) = 0} 6=∅.
Theo em 1 ensu es ha i m≤2 any nonnega i e solu ion o (7) is posi i e. Ou p incipal esul
is:
Theo em 2 Le m > 2,a > 0and A0=in {x∈Ω : a(x)=0} 6=∅. Le x0∈A0and ε > 0be
such ha B(x0,2ε)⊂A0and assume ha
aM<dm−1
m−2ε2
m−2
[q(q−1 + N−1
2)] 1
m−2
,(10)
wi h q= 2m/(m−2). Then he e exis s a dead co e o any u > 0solu ion o (7). Mo eo e ,
we ha e
Ω0={x∈Ω : u(x) = 0} ⊃ B(x0, ε).
7
P oo . Le x0∈A0and ε > 0 be such ha B(x0,2ε)⊂A0. We will build a unc ion which is
ze o in B(x0, ε)⊂Ω and bigge han he maximal solu ion o (7). Wi hou loss o gene ali y we
can suppose ha x0= 0. We conside he unc ion
Ψ(x) := 




0 i x∈B(0, ε),
(|x|−ε)q
εqi x∈B(0,2ε) B(0, ε),
1 i x∈Ω B(0,2ε).
I is clea ha Ψ ∈H1(Ω). We will show ha he unc ion u0:= KΨ sa is ies
θ[a,d,m]≤u0in Ω,
wi h an app opia e cons an K. Taking
K≥(aM
d)m(11)
we ha e u0≥θ[a,d,m]in Ω B(0,2ε) om (8). We will p o e now ha (θ[a,d,m]−u0)+= 0 in
B(0,2ε). I is su icien o p o e ha
ZB(0,2ε)|∇(θ[a,d,m]−u0)+|2≤0,(12)
because (θ[a,d,m]−u0)+= 0 on ∂B(0,2ε). Le φ∈H1
0(B(0,2ε)), φ ≥0. Then
ZB(0,2ε)∇(θ[a,d,m]−u0)·∇φ=ZB(0,2ε)∇θ[a,d,m]·∇φ−KZB(0,2ε) B(0,ε)∇Ψ·∇φ=
ZB(0,2ε)
θ1/m
[a,d,m](a(x)−dθ1/m
[a,d,m])φ+KZB(0,2ε) B(0,ε)
∆Ψ ·φ−KZ∂B(0,ε)
∂Ψ
∂n φ.
Now using ha ∂Ψ/∂n = 0 on ∂B(0, ε) and a(x) = 0 in B(0,2ε), we ind ha
ZB(0,2ε)∇(θ[a,d,m]−u0)·∇φ≤ −ZB(0,2ε)
dθ2/m
[a,d,m]φ+KZB(0,2ε) B(0,ε)
∆Ψ ·φ=
=−ZB(0,ε)
dθ2/m
[a,d,m]φ+ZB(0,2ε) B(0,ε)
(−dθ2/m
[a,d,m]+K(q
εq(|x|−ε)q−2(q−1 + N−1
|x|(|x|−ε))))φ.
Taking φ= (θ[a,d,m]−u0)+, we ob ain
ZB(0,2ε)|∇(θ[a,d,m]−u0)+|2≤
≤ZB(0,2ε) B(0,ε)
(−dθ2/m
[a,d,m]+K(q
εq(|x|−ε)q−2(q−1 + N−1
|x|(|x|−ε))))(θ[a,d,m]−u0)+.
(13)
Using (13) and no ing ha q−2 = 2q/m, o p o e (12) i is su icien o show ha
K1−2/m q
ε2(q−1 + N−1
|x|(|x|−ε)) −d≤0x∈B(0,2ε) B(0, ε).
8
We deno e
K1:= K1−2/m
ε2q(q−1) −dand K2:= K1−2/m
ε2q(N−1).
We mus p o e ha
K1+K2≤εK2
|x| o ε < |x|<2ε. (14)
Assume ha N≥2. I K1+K2≤0, (14) is i ial because K2>0. I K1+K2>0, (14) is
equi alen o
|x| ≤ εK2
K1+K2
.
This inequali y is ue in B(0,2ε) B(0, ε) i
2K1+K2≤0.(15)
I N= 1 hen K2= 0, and so (15) is equi alen o be K1≤0. I is no ha d o p o e om (10)
he exis ence o a cons an Ksa is ying (11) and (15). This inishes he p oo .
•
Rema k 3 We ema k ha he cons an
C:= C(d, ε, m, N) = dm−1
m−2ε2
m−2
[q(q−1 + N−1
2)] 1
m−2
sa is ies C↑+∞as d↑+∞o ε↑+∞. Then, i a > 0is gi en, Ω0exis s i dis la ge enough.
On he o he hand, i dis gi en, Ω0exis s i aMis su icien ly small o i A0is la ge.
Now we conside he case whe e achanges sign. Le ∈IR+and we de ine
a (x) := a+(x)− a−(x)
whe e a±(x) := max{0,±a(x)}and we suppose ha a+6≡ 0. The aim is now o s udy he
exis ence o a dead co e o solu ions o
(−∆u=a (x)u1/m −du2/m in Ω,
u= 0 on ∂Ω, (16)
wi h as a pa ame e . To p o e he main esul we need he ollowing Lemma mo i a ed by
Lemma 7 in [30].
Lemma 1 Le R > 0,
R0:= sN
2Z+∞
0
ds
[Zs
0
(Rν1/m +dν2/m)dν]1/2
9
Theo em 6 The ollowing asse ions a e ue:
1. Assume λ > 0. Then he e exis s ρ(λ)>0such ha i 0<µ<ρ(λ)any nonnega i e
solu ion o (20) has a dead co e.
2. Assume µ > 0. Then he e exis s λ0(µ)<0and ε > 0such ha i λ∈(λ0(µ), λ0(µ) + ε)
any nonnega i e solu ion o (20) has a dead co e.
3. Assume µ > 0and λ > 0. Then he e exis s c0>0such ha i c > c0any nonnega i e
solu ion o (20) has a dead co e.
P oo . Using P oposi ions 1 and 2 i is easy o p o e ha o any nonnega i e solu ion (u, ) o
(20) we ha e
u≤θ[λ+bθ1/n
[µ,n],m] ≤θ[µ−cθ1/m
[λ,m],n],(26)
and so,
Ωu,0:= {x∈Ω : u(x) = 0} ⊃ {x∈Ω : θ[λ+bθ1/n
[µ,n],m](x) = 0},
Ω ,0:= {x∈Ω : (x) = 0} ⊃ {x∈Ω : θ[µ−cθ1/m
[λ,m],n](x) = 0}.
To p o e he i s pa we will use Theo em 3 and Rema k 5. In his case
a(x) = µ−cθ1/m
[λ,m](x).
Hence aM=µand A−={x∈Ω : µ−cθ1/m
[λ,m](x)<0}.Le x0∈Ω and R > 0 be such ha
θ1/m
[λ,m](x0) = (θ1/m
[λ,m])Mand c(θ1/m
[λ,m])M−R > 0. We de ine
B(x) := cθ1/m
[λ,m](x)−R,
and so,
N(R, 1) := N(µ) = {x∈Ω : µ≤B(x)}.
I is easily seen ha i µ1≤µ2 hen
N(µ2)⊂N(µ1).(27)
Le δ > 0 be such ha 0 < δ < c(θ1/m
[λ,m])M−Rand de ine
µ0:= c(θ1/m
[λ,m])M−R−δ.
Wi h hese choices we can see ha x0∈N(µ0) and om he con inui y o B(x) he e exis s
µ0>0 such ha B(x0, µ0)⊂N(µ0). Mo eo e , om (27) we ge
B(x0, µ0)⊂N(µ0)⊂N(µ) o µ∈[0, µ0].
On he o he hand, i µ↓0 hen aM=µ↓0, and so CR↓0. Thus, he e exis s µ1>0 such
ha i 0 < µ < µ1 hen CR< µ0. Finally, we ake ρ:= min{µ0, µ1}. Thus, o µ∈(0, ρ)
CR< µ0and B(x0, µ0)⊂N(µ)
16

and so,
dis (x0, ∂N(µ) ∂Ω) ≥dis (x0, ∂B(x0, µ0) ∂Ω) = µ0> CR.
The e o e M(R, 1) 6=∅,and so he i s pa ollows om Theo em 3.
In o de o p o e he second pa le
a(x) = λ+bθ1/n
[µ,n](x)
and so,
aM=λ+b(θ1/n
[µ,n])M.
We ake
λ0=−b(θ1/n
[µ,n])M.
I is clea ha i λ > λ0 hen aM>0. Le Rand δbe such ha 0 < R < δ < −λ0. Thus, o
λ∈(λ0,−δ] he se
N(R, 1) = {x∈Ω : λ≤ −R−bθ1/n
[µ,n](x)} 6=∅.
Obse e ha i λ↓λ+
0 hen aM↓0, and so CR↓0. Now, easoning as in he abo e pa ag aph,
he esul ollows.
To p o e he las pa we ix λ, µ > 0. We ake R=c1/2and deno e by N(R, 1) := N(c). Thus,
N(c) = {x∈Ω : µ+c1/2
c≤θ1/m
[λ,m](x)}.
Obse e ha he unc ion
(c) := µ+c1/2
c
is dec easing and ends o 0 as c↑ ∞. Mo eo e , i c1≤c2 hen N(c1)⊂N(c2).Hence, o c
su icien ly la ge we ge N(c)6=∅. Finally, obse e ha CR(c)↓0 as c↑ ∞. We can comple e
he p oo by using an a gumen simila o ha used p e iously.
•
Rema k 7 We can gi e a biological in e p e a ion o he abo e esul . In he pa 1 we ix he
p eda o ’s g ow h a e. Then i he p ey’s g ow h a e is small, he p ey do no li e in all he
space Ω, i.e., he e exis some sub egions o Ωwhe e he p ey is d i en o ex inc ion by he
p eda o . On he o he hand, in pa 3 i he a e a which he p ey is consumed by he p eda o
( he pa ame e cin he se ing o he sys em (20)) is su icien ly la ge, hen he e exis some
a eas whe e he p ey do no exis .
In pa 2, om (26) obse e ha i λ≤ −b(θ1/n
[µ,n])M:= λ0 hen he e a e no semi-coexis ence
s a es o (20). Theo em 6ensu es ha (20) does no admi a coexis ence s a e i λ∈(λ0, λ0+ε).
Rema k 8 Ou esul s a e imp o emen on p e ious esul s pape s. Indeed, in [20] he au ho s
ob ained unde condi ion (25) he exis ence o a coexis ence s a e o (20) bu only in he pa icula
case 1< m =n < 2. In [7] he au ho s showed he exis ence o a semi-coexis ence s a e when λ
and µsa is y (25). I is clea ha condi ion (22) is weake han (25).
17
3.2 Compe i ion
Conside he ollowing model whose solu ions ep esen he s eady s a es o dynamical models
o compe ing popula ions





−∆u=λu1/m −u2/m −bu1/m 1/n in Ω,
−∆ =µ 1/n − 2/n −c 1/nu1/m in Ω,
u= = 0 on ∂Ω.
(28)
Conce ning he exis ence o semi-coexis ence s a es, we ha e he ollowing esul :
Theo em 7 (28) possesses a semi-coexis ence s a e i and only i λ > 0and µ > 0.
P oo . Le (u, ) be a semi-coexis ence s a e o (28). Then he maximum p inciple ensu es ha
λand µha e o be posi i e. Assume now λ, µ > 0. I is clea ha
B1:= {x∈Ω : λ−bθ1/n
[µ,n](x)>0} 6=∅, B2:= {x∈Ω : µ−cθ1/m
[λ,m](x)>0} 6=∅.(29)
We ha e o ind a couple (u, u)−( , ) o sub-supe solu ions o (28), see [12], i.e. u, u, , ∈
H2(Ω) ∩L∞(Ω), u≤uand ≤ in Ω and u≤0≤uand ≤0≤ on ∂Ω and such ha









−∆u≤λu1/m −u2/m −bu1/m 1/n in Ω,
−∆u≥λu1/m −u2/m −bu1/m 1/n in Ω,
−∆ ≤µ 1/n − 2/n −c 1/nu1/m in Ω,
−∆ ≥µ 1/n − 2/n −c 1/nu1/m in Ω.
(30)
We pick
(u, u) = (θ[λ−bθ1/n
[µ,n],m], θ[λ,m]) ( , ) = (θ[µ−cθ1/m
[λ,m],n], θ[µ,n]).
Using (29) and Theo em 1, i can be shown ha u > 0, > 0. Mo eo e , om P oposi ion 1,
u≤uand ≤ in Ω. I is no ha d o p o e ha his couple sa is ies he inequali ies in (30)
and his comple es he p oo .
•
The p oo s o he ollowing esul s a e simila o he p eda o -p ey ones, so we omi hem.
P oposi ion 4 Le (u, )be a semi-coexis ence s a e o (28). Then
u≤θ[λ,m] ≤θ[µ,n].(31)
Wi h espec o he exis ence o coexis ence s a es we ha e he ollowing esul whose p oo can
be ound in [12].
Theo em 8 Assume bc < 1and
λ > bµ and µ > cλ. (32)
Then he e exis s a leas one coexis ence s a e o (28).
18
Finally, conce ning condi ions on λand µunde which any coexis ence s a e has no a dead
co e, we ob ain:
P oposi ion 5 Assume bc < 1and ha λand µsa is y (32). Then any nonnega i e solu ion
o (28) has no a dead co e.
The ollowing esul gi es us condi ions which ensu e ha any nonnega i e solu ion o (28) has
a dead co e.
Theo em 9 The ollowing asse ions a e ue:
1. Assume λ > 0. Then he e exis s µ(λ)>0such ha i 0< µ < µ(λ)any nonnega i e
solu ion o (28) has a dead co e.
2. Assume µ > 0. Then he e exis s λ(µ)>0such ha i 0< λ < λ(µ)any nonnega i e
solu ion o (28) has a dead co e.
3. Assume λ, µ > 0and b > 0. Then he e exis s c0>0such ha i c > c0any nonnega i e
solu ion o (28) has a dead co e.
4. Assume λ, µ > 0and c > 0. Then he e exis s b0>0such ha i b > b0any nonnega i e
solu ion o (28) has a dead co e.
P oo . We will p o e he i s and hi d pa s. The o he ones ollow simila ly. We claim ha
o any nonnega i e solu ion (u, ) o (28), we ha e
u≤θ[λ−bθ1/n
[µ−cθ1/m
[λ,m],n]
,m], ≤θ[µ−cθ1/m
[λ−bθ1/n
[µ,n],m]
,n].(33)
Indeed, using P oposi ion 1 i is su icien o show ha uis subsolu ion o


−∆w= (λ−bθ1/n
[µ−cθ1/m
[λ,m],n])w1/m −w2/m in Ω,
w= 0 on ∂Ω,
which is ue i θ[µ−cθ1/m
[λ,m],n]≤ . ¿F om P oposi ions 4 and 1, i ollows ha
=θ[µ−cu1/m,n]≥θ[µ−cθ1/m
[λ,m],n].
This p o es he i s ela ion o (33). The second one ollows analogously. So, om (33) we ha e
Ωu,0={x∈Ω : u(x) = 0} ⊃ {x∈Ω : θ[λ−bθ1/n
[µ−cθ1/m
[λ,m],n]
,m](x) = 0},
Ω ,0={x∈Ω : (x) = 0} ⊃ {x∈Ω : θ[µ−cθ1/m
[λ−bθ1/n
[µ,n],m]
,n](x) = 0}.
We now p o e he i s pa . Fo ixed λ > 0, we conside
a(x) = µ−cθ1/m
[λ−bθ1/n
[µ,n],m](x),
19
and so aM=µ. As λ > 0, he se {x∈Ω : λ−bθ1/n
[µ,n](x)>0} 6=∅and hus
θ[λ−bθ1/n
[µ,n],m]>0 o µ≥0.
Le
a(x, µ) := cθ1/m
[λ−bθ1/n
[µ,n],m](x).
¿F om P oposi ion 1, i µ1≤µ2 hen
a(x, µ2)≤a(x, µ1).(34)
On he o he hand, a(x, 0) = cθ1/m
[λ,m]À0 and so he e exis s µ0>0 such ha o 0 ≤µ<µ0
we ha e
0≤µ < (a(x, µ0))M≤(a(x, µ))M.
By he con inui y o a(x, µ0) he e exis s R0>0 such ha o µ < µ0 he se
{x∈Ω : µ≤a(x, µ0)−R0} 6=∅.
Hence,
N(µ) := N(R0,1) = {x∈Ω : µ≤a(x, µ)−R0} ⊃ {x∈Ω : µ≤a(x, µ0)−R0} 6=∅.
Finally, om (34) i ollows easily ha i µ1≤µ2 hen N(µ2)⊂N(µ1).Now we can eason as
in he i s pa o Theo em 6.
We will p o e he hi d pa . Fo ixed λ, µ > 0 and b > 0, we conside he same unc ion a(x)
as abo e. In his case, we ha e
N(R, 1) := N(c) = {x∈Ω : µ+R
c≤θ1/m
[λ−bθ1/n
[µ,n],m](x)},
and again a guing as o he second pa o Theo em 6 he p oo ollows.
•
Rema k 9 1. Theo em 9has a biological in e p e a ion simila o Theo em 6gi en in Re-
ma k 7.
2. The exis ence o semi-coexis ence s a es o (28) has been s udied in [7]. The au ho s ob-
ained a semi-coexis ence s a e unde he condi ion (32), which is s onge han λ, µ > 0.
I seems ha he exis ence o coexis ence s a es o (32) has no been analyzed p e iously.
3.3 Symbiosis
In his case we conside he coope a i e sys em





−∆u=λu1/m −u2/m +bu1/m 1/n in Ω,
−∆ =µ 1/n − 2/n +c 1/nu1/m in Ω,
u= = 0 on ∂Ω.
(35)
Ou main esul abou he exis ence o semi-coexis ence s a es is he ollowing one:
20
Theo em 10 Assume bc < 1and ha he se s
P1:= {x∈Ω : µ+cθ1/m
[λ,m](x)>0} 6=∅P2:= {x∈Ω : λ+bθ1/n
[µ,n](x)>0} 6=∅.
Then he e exis s a leas one semi-coexis ence s a e o (35).
Rema k 10 An obse a ion abou he way in which he abo e condi ions should be in e p e ed
is app op ia e. I o example λ≤0 hen θ[λ,m]= 0, and so P16=∅is equi alen o µ > 0.
P oo . Again we use he sub-supe solu ion me hod o sys ems. In his case a couple (u, u)−( , )
is sub-supe solu ion o (35) i , see [12], u, u, , ∈H2(Ω) ∩L∞(Ω), u≤uand ≤ in Ω and
u≤0≤uand ≤0≤ on ∂Ω and such ha









−∆u≤λu1/m −u2/m +bu1/m 1/n in Ω,
−∆u≥λu1/m −u2/m +bu1/m 1/n in Ω,
−∆ ≤µ 1/n − 2/n +c 1/nu1/m in Ω,
−∆ ≥µ 1/n − 2/n +c 1/nu1/m in Ω.
(36)
We conside wo cases:
1. Assume λ≤0. As we ha e men ioned abo e, he condi ion P16=∅is equi alen o µ > 0.
We ake
(u, u) = (θ[λ+bθ1/n
[µ,n],m], M) ( , ) = (θ[µ,n], N),
wi h M, N > 0 cons an s s ill o be chosen. Since P26=∅ hen u > 0.
I is easy o show ha Mand Nsa is y (36) p o ided ha
λ−M1/m +bN1/n ≤0, µ −N1/n +cM1/m ≤0.(37)
Since bc < 1 he e exis Mand Nsa is ying (37). Thus he p oo o case 1 is comple e.
2. Assume λ > 0. In his case, P2= Ω. Now we pick
(u, u) = (θ[λ,m], M) ( , ) = (θ[µ+cθ1/m
[λ,m],n], N),
wi h Mand Nas abo e. This comple es he p oo .
•
Again he ollowing esul s do no need any p oo .
P oposi ion 6 Assume bc < 1and le (u, )be a nonnega i e solu ion o (35). Then
θ[λ,m]≤u≤µλ+bµ
1−bc ¶m
θ[µ,n]≤ ≤µµ+cλ
1−bc ¶n
.(38)
As an immedia e consequence o P oposi ion 6, we ha e
21

Co olla y 5 Assume bc < 1and ei he λ+bµ ≤0o µ+cλ ≤0. Then (35) does no admi a
semi-coexis ence s a e.
We ob ain he ollowing esul on coexis ence s a es
Theo em 11 Assume bc < 1and
(λ, µ)∈IR+×IR+ {(0,0)}.(39)
Then (35) possesses a leas one coexis ence s a e.
P oo . Assume λ, µ > 0. We ake
(u, u) = (θ[λ,m], M) ( , ) = (θ[µ,n], N)
wi h Mand Nsa is ying (37). I is no di icul o show ha his couple is a sub-supe solu ion
o (35). On he o he hand, i o example λ= 0 and µ > 0 we pick
(u, u) = (θ[bθ1/n
[µ,n],m], M) ( , ) = (θ[µ,n], N).
In his case, Mand Nha e o sa is y he ollowing inequali ies
−M1/m +bN1/n ≤0, µ −N1/n +cM1/m ≤0.
We can eason analogously in he emaining case. This concludes he p oo .
•
The ollowing esul can be p o ed simila ly o P oposi ion 3.
P oposi ion 7 Assume (39). Then any nonnega i e solu ion o (35) is posi i e, i.e., i has no
a dead co e.
The ollowing esul says ha he se o (λ, µ)∈IR2whe e (35) has a leas a semi-coexis ence
s a e is connec ed.
Theo em 12 Assume bc < 1. Le us deno e
Γ := {(λ, µ)∈IR2: (35) has a leas a semi-coexis ence s a e in (λ, µ)}.
Then Γis connec ed.
P oo . Le (u0, 0) be a semi-coexis ence s a e in (λ0, µ0). We ix λ0. Le µ∈[µ0,∞) and de ine
(u, u) = (u0, M) ( , ) = ( 0, N)
wi h Mand Nsa is ying (37). I is no ha d o show ha his couple is a sub-supe solu ion o
(35) in (λ0, µ). This comple es he p oo .
•
Rema k 11 To ou knowledge, (35) has been only s udied p e iously in [7]. In his wo k he
au ho s ob ained he exis ence o a semi-coexis ence s a e unde he condi ion λ > 0and µ > 0.
F om Theo em 10,(35) possesses a semi-coexis ence s a e e en when λo µis nega i e.
Acknowledgemen s. We a e g a e ul o P o esso s J. He n´andez and J. L. G´amez o help ul
commen s. The au ho s hank o DGICYT and CICYT o Spain o esea ch suppo unde
g an DGICYT PB95-1242 and MAR98-0486, espec i ely.
22
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24