Positive solutions for predator-prey equations with nonlinear diffusion rates

POSITIVE SOLUTIONS FOR PREDATOR-PREY EQUATIONS WITH NONLINEAR DIFFUSION RATES

IN KVUNG AHN

1.

Introduction

In this paper, we will investigate the existence of positive solutions to the predator-prey interacting system

-

_{~(x, u)~u}

= uf(x,

u,

v) -

~(x,v)~v

= vg(x,u,v)

au an +

^IW

= 0 on an an av ⁺

^O'V

= o.

in n

(1)

in a bounded region n in Rn with smooth boundary, where

~

and

~

are strictly positive functions, serving as nonlinear diffusion rates, and

^Ko, 0'

> 0 are constants. Assume that the growth rates f,

9

are Cl monotone functions. The variables

u,

v may represent the pop- ulation densities of the interacting species in problems from ecology, microbiology, etc.

In [12], the solutions of the equations

{

-~u: ^uM(x,u,v)

-~v

- vN(x,

u,

v) in n

under the boundary conditions

~:

=

~:

= 0 are investigated in the competition and symbiosis cases, using the monotone-iteration scheme.

For the predator-prey case, the variational approach was used in [4] for the first time.

Received June 17, 1993. Revised November 29,1993.

AMS Subject Classification 35J60.

This work was partially supported by GARC-KOSEF.

In [5], [6] and [7], necessary and sufficient conditions for the existence of positive solutions of the elliptic system

{

- dldu = uM(u,v) - d

2

dv = vN(u, v)

u=v=O on 80

with constant diffusion rates have been established for all the possible cases of monotonicities associated with the functions M and N. For those works, index theory ([5], [7]) and decomposed operators ([6]) have been employed to prove the existence of positive solutions.

Therefore, the problems of the existence of positive solutions for the nonlinear interacting systems with constant diffusion rates are to a large degree solved. However, in many chemical and biological systems, the diffusion rates indeed depend on the densities u and v. Thus we need to stu.dy the readion-diffusion systemS .with no:iilinear diffusion rate from this point of view.

To attack our problem, we first give some a priori estimates on solutions to the system (1) to have the compactness of the correspond- ing operators for the nonlinear elliptic equations. We then handle the linearization to equations.

In section 2, we provide a sequence of lemmas which will be used to prove the results in section 3. In section 3, we will give suffi- cient and necessary conditions for existence of positive solutions of (1) for the predator-prey interaction. The existence of positive solutions can be characterized by the spectral· property of a certain operator of Schrodinger type.

2. Preliminaries

In this paper we will consider problems in the space X = C(fi), where n is a bounded region in Rn and let r(T) denote the spectral radius of a linear operator T.

Let I = I(x, e). Then I

E

F ifand only if I E C(fi

X

R+) and (Fl) I E

Cl

in e, fe(x,e) < ° ^{in n} x R+, and for some N

E

R+, Ife(x,e)1 :5 N where (x,e)

E

n ^x [0, eo].

(F2) I(x, 0) > ° ^{and I(x, e) < 0, where (x,e) En x (eo, 00) for some}

constant eo > 0.

(F9)

f(x,~)

is concave down on the set of (x, 0 where f(x, 0 < O.

Let <p = <p(x,O. Then <p

E

G if and only if <p

E

C(fi

^X

R+) and <p is Cl-function in

~

which in addition satisfies

(GI)

<p(x,~) ~

6 > 0 for some constant 6 and

~ E

R+, x

E

O.

( G

2) <p is nondecreasing and concave down in

_~ E

R +.

LEMMA 1.

Let f

E

F and

<p E

G. Then 'P J~x,u~ is decreasing in u

^X,B for x E

O.

Proof. We consider bounded regions OJ,

j =

1,2,3, 0

1 =

{x E

o : f(x,ut}

~

0 and f(X,U2) > O},

O2

= ^{x

^EO:

^f(x,ut} <

o ^{and f(x,u2)}

~ A}, 0₃

= {x

EO;

f(x,ut} < 0 and !(X,U2) 50}.

Then 0 = 0

1

U 0

2

U03.

For

x E

0

₁

or

x E

O

_{2 ,}

it is easy to see the monotonicity of ~.

Suppose x

EO.

Note that 'P(x,ttt> < 'P(X,U2) and

~

< J(X,U2) by

3 ttl - tt2 UI - tt2

the assumptions (F3) and (G2). Since f(x, ut} s ^{f(x, U2) 5 0, one}

must have J(:c,ud . 'P(X,U2) <

~

. 'P(x,u d . Therefore we have that

ttl U2 - tt2 ttl

for x

E

0

_3, ~

_'P(x,ttd -

~_~

= _'P(x,ud

^{X UI UI X U2 tt2}

^<

^O.

^{Thus for}

UI

'P X,U2

tt2 -

x

E

0, 'P X,tt J«x,tt» is decreasing in u.

LEMMA

2. Let P > 0 be a constant. Assume

<p E

G. Also let

o 1= h

_~

0, h E CO(fi) where

0:'

E (0,1). Consider

{

- <p(x,

u)~u

+ Pu

= h

on ou ⁺

^IW

= ^{0 on 00. (2)}

Then (2) has

a

unique positive solution

u

E C2,O(n). Moreover, the solution operator S such that u = Sh is compact in K or C(fi), where K is the positive cone ofC(n).

Proof. The positiveness of solutions to (2) for

h _~

0 follows from the strong maximum principle. First we show that the nonnegative solution to (2) is unique when 0 ShE CO(fi). Let u and

^'t!

be two distinct nonnegative solutions of (2). Without loss of generality, let minxEo(u(x)-v(x)) < O. Let u(xo)-v(xo) = minxEo(u(x)-v(x)) < O.

Assume Xo

'E

00. Then by the minimality of u - v at xo, we have

a(tt-a~(xo)

^SO

and K[U(XO) - v(xo)] <

O.

Thus the boundary condition

(4) becomes [8uJ:o) - 8vJ:o)]+I\;[u(xo)-v(xo)] < 0, which is a contradi~tion.

Thus Xo (j. an. Since u and v are solutions of (2), we have

- <p(x, u)<p(x, v

)~(

u - v) + P(u<p(x, v) - Vlp(X, u» (3)

=h(x)(<pex, v) - cp(x, u))

Observe that at Xo EO, -cp(xo, u(xo»<p(xo,

v(xo»~(u(xo)-v(xo»

s 0 by the minimizing property of Xo. Also we have

p.

[u(xo)cp(xo, u(xo»- v(xo)cp(xo, u(xo»] < 0 since the nondecreasing function <p > 0 is con- cave down. Hence the left side of (3) is negative. Since v(xo) > u(xo) and h(xo)

~

0, the right side of (3) is nonnegative, which is a contra- diction. Therefore we must have u = v.

Next we shall prove the existence of a solution. First we can show the a priori bound in a space C

²

,Q(fi) of every solution to (2), where a E (0,1) and then find a fixed point of the following equation

{

-~u ⁼ :0:~)

~: + I\;U = ^{0 on} ao.

Let u be a solution of equation (2) for x

E

0, i.e., u is a fixed point of equation (4). Denote Green's function under the Robin boundary condition

:~

+

^1\;.

⁼ 0 by G

R.

Then

( (h-PU(Y»)

u(x) = ln GR(X,y) <p(y,u(y» dy. ⁽⁵⁾ Since IIPull oo s IIhll oo ^by a general maximum principle, we can esti- mate:

LIGR h -<pPu I ⁽⁶⁾

SIlGRIILn'n-t1lh - PUIlLn .8-

¹

< K1211hlloo8-1 = K

2

l1hll oo.

{I ^aG

^R

^{h - Pu} I ⁽⁷⁾

ln

^aXj

^cp

s :"'111 h - PUIlLm L r~ ^{< 211 h lloo 8- 1 N} =Kllhll~

where N ⁼ J

1!!.::.!lm.

1 and r = IIx - yll. Thus (6), (7) imply

r m-l

(8) So applying the elliptic regularity to (4) and the fact lIV'ull

oo :::;

Mllhll

oo ,

we have u E W

²

,m(S1) where m > n and by (8)

(9) for some constant C. Since m > n, the Sobolev imbedding theorem implies

(10) where some

a

E (0,1). On the other hand, the Schauder estimate ([2]) gives

lIullc2,o :::;

C1

(n, a)(lIull

oo

+ 6-

¹

·lIh - Pullc

^{o ).} (11)

Therefore lIullc2... is also bounded.

Let v E C

²

,O(0) and define T : C

²

,O(0) -. C

²

,O(0) such that

u

= Tv is a solution of (4). Then T is continuous, compact and bounded.

Therefore the Schauder fixed point theorem provides a fixed point u such that Tu = u.

The compactness of the solution operator S in C(O) such that

u

=

Sh

follows from (9), (10) and (11) by the Ascoli-Arula theorem.

COROLLARY.

Let P > ° ^and

^T

^> ° ^{be constants. Suppose a( x)}

^E

C

¹

(0) and a(x)

~

6

0

> 0, Consider for a E (0,1),

{

~a(x)~u

+ Pu = h(x),

an ⁺

^{TU =}

^{0, on an. (12)}

Then (12) has a unique solution u in C

²

(0) and the solution operator T such that u = Th is a compact operator in X = C(O).

The following is well-known. We record this to use in the proof of

other results.

LEMMA 3. Suppose a E C 1 (fi) and a(x) ;::: Co > 0 and b(x) E

£=(n). Let

^T

> 0 be a constant. Then there exists u > 0, u E C2(fi), such that for a unique Al > 0

{

-

a(x)~u

+ ^{b(x)u = A1U} an au +

^{TU =}

0 on an.

Moreover, Al is increasing in a( x) and a ratio :~:~.

(13)

Let X be a Banach space and let F be a strongly positive nonlinear compact operator on X such that F(O)

=

O.

LEMMA 4. Assume F'(O) exists with r(F'(O» > 1. H there is no p E (0,1} in any neighborhood of whi.ch. the equation u = pF'IJ,h.C!S a solution u as lIull _

^00,

^{then F has} a positive fixed point u such that Fu

=

u in the positive cone K of X.

Proof. See Theorem 13.2 in [1].

LEMMA 5. Let T be a compact positive linear operator on an or- dered Banacb. space. Let u > 0 be a positive element.

(i) HTu > u, then r(T) > 1 (ii) H Tu < u, then r(T) < 1 (iii) HTu = u, then r(T) = 1.

Proof. See Lemma 2.3 in [5].

LEMMA 6. Let a(x)

~

Co > 0 and b(x) E £=(n). Also let P be positive constant such that P + ^b(x) > 0 for x E n. ^Then

(i)

A1(a(x)~

+ b(x)) > 0 {::} r[(

-a(x)~

+ P)-l(P + b(x »] > 1 (ii) A1(a(x)6. + ^b(x» < 0 {::} r[( -a(x)6. + ^P)-l(P + ^b(x»} < 1 (iii)

A1(a(x)~

+ ^b(x)) = ^{0 {::} r[(}

-a(x)~

+ ^P)-l(P + ^b(x»] = ¹

where Al is the first eigenvalue under Robin boundary condition.

Proof. Let <jJ > 0 be the eigenfunction corresponding to the eigen- value A1(a(x)6.+b(x)). Then

("""",a(x)~+P)<jJ

= (P+b(x

»<jJ-A1(a(x)~

+b(x))<jJ. So one can see that

(-a(x)~

+ P)<jJ >, =, < (P + b(x))<jJ de- pending on the sign of

A1(a(x)~+b(x)).

Let T:=

(-a(x)~+p)-l(p+

b(x». Then T is a positive compact operator under Robin boundary

condition in C(n) by Corollary. Therefore one may apply Lemma 5 to get the results.

Consider the following equation,

{

- lp(x,

u)~u

= uf(x, u)

au an +

^{KU =}

⁰ on an.

in 0.

(14)

Next we shalllinearize the equation (14) at

u

= O. Use Lemma 2 to define the solution operator S in C(n) by Su

=

ii, where ii is the unique solution of

{

- lp(x,

ii)~ii

+ Pii

=

uf(x, u) + Pu

ail -

0

~{"\

an ⁺

^{KU =}

^on

^VH.

⁽¹⁵⁾

Also we define the solution operator SL of linearization by SLW = v, where

v

is the unique solution of

{

-

lp(x,o)~v

+ Pv = wf(x,O) + Pw

an

Qv

+

^KV

⁼ ° ^{on an. (16)}

LEMMA 7. The operator S is FTechet differentiable at u = 0, and S'(O) = SL.

Proof. Let ii

=

Su be the unique solution of (15) and v be the

solution to (16). Assume that Ilull

oo

is small. Then from (15) and

(16), we have

It is easy to see that lIulloo

⁼

O(lIulloo), and IIhlloo

=

o(lIulloo) when u is small. Thus lIu - vlloo = O(lIhlloo) ⁼ o(lIulloo).

Let K = C(fi)+ be the positive cone of the ordered Banach space C(fi). We define the ordered interval [[u}, U2]]

:=

{u

E

C(fi) :

UI ~

U

~

U2 for

UI,

U2

E

C(fin. In the next two lemmas, AI(A) denotes the first eigenvalue of an operator A under the boundary condition

~: +

^KU =

0.

LEMMA

8. Let I E F and e.p E G.

(i) If AI(e.p(X,O)d + l(x,O» > 0, then the equations (14) have

a

unique positive solution in C2(fi).

(ii) If AI(e.p(X,O)d+ l(x,O»

_~

0, then

^U

== ° is the only nonnegative solution of (1.1,).

Proof (i) Suppose AI(e.p(X,O)d + I(x, 0» > 0.

If

u is a positive solution of (14), then °

^~

^{u <}

^Co

by the general maximum principle.

Choose P > 2max{suPfix[o,co]l/(x,e)l,co L}. Let [[0, CO]] denote the order interval in C(fi). We define an operator A : C(fi)

^-+

C(fi) by Au

=

S( ul(x, u)+Pu) where S is again defined as the solution operator of the equation (2). Then A is an increasing strongly positive compact operator from [[0, CO]] to C(fi) by Lemma 2. Note that u is a positive solution of (14) if and only if

u

is a fixed point of the operator A.

It

is easy to see that u ^{== CO} is an upper solution of the equation of (14).

So we have A'(u)

~

u. Also note that u == ° is a solution of (14) and Lemma 7 implies that A'(u)

=

A'(O)

=

(-e.p(x, O)d+P)-I[f(X, O)+P].

Thus we have r(A'(O» > 1 by Lemma 6. Now applying Theorem 7.6 in [1], we have a maximal solution u in [[0, co]].

(ii) Let u be a positive solution of (14). Then we have (e.p( x, u)d + I(x,u»u

=

0. This implies AI[e.p(X,u)d + I(x,u)]

=

0. Then since

I

E

F and

g E

G, f~(~» is decreasing in u by Lemma

1.

So by the last part of Lemma 3, we have AI[e.p(X,O)d+ l(x,O)] > AI[e.p(X,u)d+

I(x, u)]

=

0, which is a contradiction.

Next we show that the above solution is unique. Suppose

UI

and U2 are two positive solutions of (14). Let u be a maximal solution of (14).

Then

U ~ ^UI

and

U ~ ^U2.

IT we can show u =

^UI

and

U

=

^U2,

then

Ut

= ^U2. Thus it suffices to show that any other positive solution

Ut

must coincide with ii.. Since ii. and Ul are solutions of (14), we have .

l

_[-Ul~U

- - +

U~Ul]

= 1- UUl ^[!(X,ii.) ( _) - ^!(x,Ud] ( )

o

0 ~

x,u

~

X,Ul (18)

The left integral of (18) is

fo [-Ul~ii. ⁺ ii.~Ul] ⁼ foo [-Ul: ^{+ ii. ';;: ] =} foo [UlK.ii. - ii.K.Ul] = o.

Since ~ is decreasing in

^U

by Lemma 1 , the right side of (18) is nonpositive. By the continuity of! and

~ g,

we must have ii.

=

Ul'

The following lemma is a generalization of Lemma 4 [6]. Here the diffusion rate is nonlinear. According to Lemma 8, the equation (14) has a unique positive solution. We denote it by u<p,f. Let u<pn,fn be the unique positive solution of

{

-

~n(x, u)~u

= u!n(x, u)

- au an + K.U

=

0 on an.

(19)

LEMMA 9.

Assume!

E

F and

~E

G.

(i)

(~,

f)

1-+

u<p,! is a continuous mapping ofG x F

-+

cl,Q(n x R+) for some a

E

(0,1).

(ii) H!:; ~ £; :t: !:;, ^{for x}

^E

n, then either

U<Pl,fl

> U<P2,f2 or

U<Pl,fl

== U<P2,f2 == O.

Proof. (i) The argument is similar to that in Lemma 4 [6]. Here we just give a simple modification for the uniform boundedness of {u<pn,fn}' Note that since fn

-+

f and

~n -+ ~,

there exists K > 0 such that

lI u<pn,fnfn(x,

u<pn,!n)

+ PU<pn,fn 1100 :5 K

Therefore, as in the proof of Lemma 2 with h _n

=

u<pn,fnf(x,u<pn,fn) +

PU<pn,fn' we have

lIu<pn,fnllw2.m(0)

:5 N by (9), and so there exists a

subsequence {u<pn,fn} such that

u<pn,!n ~

U in W

²,m(n).

Also we have

by the Sobolev imbedding theorem,

lIu<pn,fnIIcl, .. (o)

:5 M where M is

a positive constant. Thus there exists a subsequence of

{u<pn,fn},

say

{U'Pn,fn}

again, such that

u'Pn,fn -+ it

in Cl,a(fi) for some a E (0,1).

As in the proof of Lemma 2, we conclude that

it

is a solution. Then the positiveness of

u'Pn,fn

implies

it

> 0. So by the uniqueness of solution,

it

=

^u'P,f.

^Therefore

^{u'Pn,fn -+ u'P,f}

^{in Cl,a(fi).}

(ii) Suppose .h.

_'Pl

> h...

_'P2

If we assume .xl(CPl(X,O)~ + Il(X, 0» <

_-

0, then by Lemma 8,

U'Pl,ft

==

u'P2,h

== 0. SQ suppose

.xl(CPl(X,O)~

+

f

¹

^{(0» x, >.} ° ^{Th en}

^U'Pl,h

^{> .} ° ^{S· lnce}

^-uU

^{A =}

^{U'Pl(Z,U)ft(Z,u)}

^>

^{- U'P2 Z,U 'h z,u)}

u'P2,h

is a lower solution of (14). Also note that

U

=

Co

is an upper solution. Thus there exists a solution

'it

of (14) such that

u'P2,h :::;

'it :::; Co.

By the uniqueness of positive solution of the equation (14),

'it

=

u'P1>ft.

So we have

u'P1>ft ;::: u'P2,h.

Apply the strong maximal principle to get

U'Pl,fl

>

u'P2,h.

3. Existence Theorem

for u,v > 0 for u,v > 0 on ail

!,,(x,U,v) < °

g,,(x,u,v) < °

In this section, we consider the system (1):

-CP(x,u)~u

= u!(x,u,v)

-~(x,v)~v

=vg(x,u,v)

au \

an ^{+KU =0} an av ^{+uv = 0}

For the predator-prey interaction, we make the following assump- tions on the system.

(HI) !,

9

E Cl(fi

X

R+

x

R+) satisfy

{

!u(x,u,v) < 0 gu(x,u,v) > 0

Moreover, lv, gu f. ^{0 and all} partial derivatives are uniformly bounded on fi

X

R+

X

R+.

(H2) There exist positive constants Cl, C

₂

such that

{

l(x,u,O) < 0 for

^U

> Cl g( x, 0, v) < 0 for v > C

2

g(x, Cl, C

2 )

< 0

(20) (H3) Let cp(x,u), t/J(x,v) E C(n x R+) and 'P(x,'), t/J(x,') E G.

The assumption (H!) describes how these species u, v interact with each other, while the assumption (H2) indicates that the model under consideration is logistic.

Let Al,.. (A), Al,a(A) denote the first eigenvalues of an operator A under the boundary conditions :~ +

^K'

= ° ^{and :~ + a· = 0, respec-}

tively.

By Lemma

8,

if Al, ..('P(x,0)6 + j(x,O,O)) > ° ^{where j}

^E

^{F and}

'P E

G, then the equation

{

-

cp(x,u)~u

= uj(x,u,O)

au an +

^{KU =}

° ^{on an}

has a unique positive solution. Denote this positive solution by

^UQ.

Similarly, if

Al,a(t/J(X,O)~+g(x,O,O))

> ° ^{where 9}

^E

^{F and t/J}

^E

^G,

then

{

- t/J(x,v)6v

=

vg(x,O,v)

an av + av

=

° ^{on an}

has a unique positive solution

^VQ.

According to the Lemma 9, we may define the operator T : C(n)

---+

C(n) as follows. For v

E

C(n), by Lemma 8, u

=

Tv is the unique positive solution to the equation

{

- 'P(x,

u)~u

= uj(x, u, v)

an au ⁺

^{KU =}

° ^{on an}

if Al,..('P(x,0)6 + ^j(x,O,v)) > ° ^{where j}

^E

^{F and cp}

^E

^G.

REMARK.

By Lemma 9, it is easy to see that T is a continuous operator and T is strictly monotone. In case that u is a prey for v, T is decreasing in

v.

Replacing u by Tv in the other equation, we have

{

-

t/J(x,v)~v

= vg(x,Tv,v)

- - + _an av ^av = ° ^{on an}

According to Lemma 2, we define the operator A : C(n)

-+

C(n) by considering the equation

{

- 'l/J(x,w)t:.w + Pw

=

vg(x,Tv,v) + Pv

aw an +

^qW

⁼ ° ^{on an. (21)}

We denote it by w = Av = (-'l/J(x,·)t:.· +p.)-l[vg(X, Tv, v) + Pv]

where P is constant. Note that the operator A has a fixed point

v

in the positive cone K if and only if (Tv, v) is a nonnegative solution of (1).

THEOREM

1. Assume that (H1)-(H3) hold. If >"l,D'('l/J(X,O)t:. + g(x, uo,O» > ^{0, then} A has

a

positive fixed point in K.

Proof To show this, we apply Lemina 4. Denote Ae ⁼ 8A. IT Ae(ve)

=

Ve, 8

E

(0,1], ve

E

K, then we have

-'l/J(x, ~)t:.ve

⁼

8veg(x,Tve,ve) + ⁽⁸ -l)Pve. (22) Suppose ve t=

^0.

^{Let ve( xo) = maxxEo ve(x) >} ° ^{for some}

^{Xo E}

^n.

Then

Xo

must be in n by the boundary condition of (21), and at

Xo

the left side of (22) is nonnegative and (8 - l)Pve(xo)

~

0. Thus we must have g( Xo, Tve( xo), ve(xo» 2: 0. Suppose u, v are prey and predator, respectively. Since g( x, Tv, v)

~

g( x , Cb v) in v E K where Cl is a priori bound for u, we have ° ^S g(xo, Tve(xo), ve(xo» S g(xo, Cb ve(xo». Thus by the assumption (H2), ve(xo) S C

2 •

Thus there is an a priori bound for the positive fixed points of Ae. Next observe A'(O)

=

(-'l/J(x, O)t:. + P)-l(g(x, Uo, 0) + P) by Lemma 7, where (-'l/J(x,O)t:. + P)-l is a linear operator under boundary con- dition :~ +

^q. =

0. Since >"l,D'('l/J(X,O)t:. + g(x,uo,O» > 0, Lemma 6 implies r(A'(O» > 1. Note that A is strongly positive compact opera- tor from C(n) to C(n) by Lemma 2. Thus A has a positive fixed point v in K by Lemma 4.

THEOREM

2. Suppose the conditions (H1)-(H3) hold.

(a) If >"l,K(cp(X,O)t:. + !(x,O,O» SO, then (1) has no positive solu-

tion.

(b) Suppose

A1,u(1/1(X,0)~

+ g(x,O,O» > 0. (therefore there exists a solution (0, vo) with Vo > 0.) Then the system (1) has a positive solution if and only if Al,,,(CP(X,

o)~

+ f(x, 0, vo» > 0.

(c) Suppose

A1,u(1/1(X,0)~+g(x,0,0»:$

0. Then the system (1) has a positive solution if and only if Al,,,(CP(X,

O)~

+ f(x, 0, 0» > ° ^(thus

there must be a solution (uo,O) with Uo > 0) and

A1,.,.(1/1(X,0)~

+ g(x, Uo, 0» > 0.

Proof. (a) Suppose (u, v) is a positive solution of the system (1).

Then u t ° ^{and (cp( x,}

^u)~

^{+ f( x, u, v»u =} O. As in the proof of Lemma 8 (ii), Lemma 1 and Lemma 3 implies

A1,,,(CP(X,0)~

+ f(x,O,O» >

A1,,,( cp( x,

u)~

+ f(x, u, 0» > Al,,,(cp(x,

u)~

+ f(x, u, v») = 0, a contra- diction. Thus the system has no positive solution.

(b) (<=) Since

A1,.,.(1/1(X,0)~+g(x,uo,0»

>

A1,u(1/1(X,0)~+g(x,0,

0» > 0, A has a positive fixed point v in K by Theorem 1. Note that u

=

Tv > ° ^{because if Tv ==} 0, then by uniqueness, Vo

=

v.

But

Al,,,(CP(X,O)~

+ f(x,O,vo» > ° ^{implies Tv = Tvo > 0, a con-}

tradiction. Therefore (Tv, v) is a positive solution of the system (1).

(=}) Let (u,v) be the positive solution of (1). Since g(x,u,v)

~

g(x,O,v), we have

-1/1(x,vo)~vo:$

vOg(x,u,vo). So Vo is a lower so- lution of -1/1(x,

v)~v

= vg(x, u, v). Thus Vo

:$

v. So we have ° ⁼

A1,,,(CP(X,

u)~

+ f(x, u, v» < Al,,,(CP(X,

O)~

+ f(x, 0, v») < Al,,,(CP(X, 0)

~

+ f(x, 0, vo». Therefore A1,,,(cp(x,

O)~

+ f(x, 0, vo» > 0.

(c) Again, since A1,.,.( 1/1(x,

O)~

+ g(x, Uo, 0» > 0, there exists a fixed point v of A by Theorem 1. Then u = Tv > ° ^{since if Tv == 0,}

then v = Vo. Thus, by using Lemma 1 and Lemma 3, we have

°

⁼ A1,u(1/1(X,v)~

+ g(x,O,v» :::;

A1,.,.(1/1(X,0)~

+ g(x,O,O» :::; 0, a contradiction. For the necessity, let u be a positive solution with v = v of the system (1). If v = 0, then u ⁼ Uo. If v > 0, then Uo > u by the monotonicity of T. (See Remark.) Thus we have uo

~

u > O.

If

A1,,,(CP(X,0)~

+ f(x,O,O» :::; 0, we have a contradiction by (a). If A1,u( 1/1( x,

O)~

+ g( x, Uo, 0»

:$

0, then °

⁼

^{A1,u( 1/1( x,}

^v)~

^{+ g( x,}

^tt,

^{v» <}

A1,u(1/1(x, 0) + g(x, u, 0» < A1,u(

~(x,

0) + g(x, Uo, 0» :::; 0, a contradic- tion again.

References

1. H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces,SIAM Rev. 18 (1976), 620-709.

2. D. Gilharg and N. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, New York (1983).

3. J.A. Goldstein and C-Y Lin, Singular parabolic boundary value problems in one space dimension, J.of Differential Equations 68 (1987),429-443.

4.C. Keller andR. Lui, Ezistence of steady-state solutions to predator-prey equa- tions in a heterogeneous environment,J. Math. Anal. Appl. 123 (1987),306-326.

5. L. Li, Coexistence theorems of steady-state for predator-prey interacting sys- tems, Trans. Amer. Math. Soc. 305 (1988), 143-166.

6. L. Li and A. Ghoreishi, 281-295, J. Appl. Anal. 14(4) (1991).

7. L. Li and R. Logan, Positive solutions to general elliptic competition models, J.of Diff.&;Integ Eqs. 4 (1991), 817-834.

8. C.·Y. Lin, Degenerate nonlinear parabolic boundary value problems, Nonlinear Analy. Theory, Methods&;Appl. 13(11) (1989), 1303-1315.

9. G. F. Roach, Green's junctions, Van Nostrand Reinhold Company (1970).

10. J. Smoller,Shock waves and reaction-diffusion equations, Springer-Verlag, New York (1983).

11.J. Wloka,. Partial differential equations, Cambridge University Press (1987).

12.KoZygourakisand R. Aris, Weakly coupled systems of nonlinear elliptic bound- ary value problems, Nonlinear Anal. 6(6) (1982), 555-569.

Department of Mathematics Korea University

Chochiwon 339-700, Korea