monotone schemes for semilinear elliptic systems related to ecology

Math. Meth. in the Appl. Sci. 4 (1982) 272 - 285 A M S subject classification: Primary 35 J 65,92A 17

Monotone Schemes for Semilinear Elliptic Systems Related to Ecology*)

A. Leung, Cincinatti

Communicated by P. H . Rabinowitz

Semilinear elliptic systems of partial differential equations related to ecology are studied, with Dirichlet boundary conditions. Monotone sequences of functions which satisfy scalar equations are constructed so that they will converge to upper and lower bounds for the solutions of the systems. In case a related system has a unique positive solution, then these sequences will converge to the solution of the original system. Applications of the monotone sequences to uniqueness and stability are also given.

1 Introduction

The first part of this article considers the boundary value problem

Au + u(a - bu - cu) = 0 Au + u(e +fu - go) = 0 u = u = o on 8 9

in 9 (Pl)

where A = (a2/&$). Here Q, b, c, e,J g are positive constants, with a >Il,

e > Itl, where Itl > 0 is the principle eigenvalue for the problem A w + 1 w = 0 in 9, w = 0 on 6 9. (The principle eigenfunction will be denoted by o (x)) . The system studies spatial equilibrium for prey-predator interaction of Volterra- Lotka type (cf. e.g. [l], [3], [ 5 ] , [6] , [lo]). Note here that we have assumed positive intrinsic growth rate e for the predator u. This, of course, can arise if the predator has food supplies other than the prey u. We will further consider the case of more than two interacting species in the section 3. We will construct monotone sequences of functions which will converge to upper and lower bounds for all positive solutions of (Pl). In case a related system has a unique positive solution, then these sequences will converge to the solution of (pi). This constructive scheme has numerical as well as analytical contribution to the study of our problem.

*) This research was partially supported by a grant from the National Science Foundation, MCS 80-01851.

i= 1

01704214/82/02 0272- 14 S 02.80/0 0 1982 B. G. Teubner Stuttgart

Monotone Schemes for Semilinear Elliptic Systems Related to Ecology 273

Note also that the coefficient of uv in the two equations of (Pl) have opposite signs. This makes the constructive scheme difficult as indicated in [7], p. 999. Moreover, our scheme can be generalized to systems of order higher than two, as indicated in the section 3. Applications to uniqueness and stability is given in section 4.

In order to insure the existence of positive solutions we will make the following further assumptions in section 2:

c f < g b

Such situations will happen when the intrinsic growth rate, a, of the prey is large and intraspecies interaction is relatively large compared with interspecies interaction.

To fm ideas, we will assume 9 is a bounded domain in R", n 2 1 . H2+' (3 ) , 0 < I < 1, denotes the Banach space of all real-valued functions u continuous in d with all first and second derivatives also continuous in 8, and with finite value for the norm lul:+'(as described in [lo], p. 159). 69denotes the boundary of 9, and we assume that 69 E H2+' (see [lo]). All solutions to our equations will mean solutions in ~ ~ + ' ( k d ) .

2 Monotone Scheme

Before our construction of monotone sequences, we need the following two lemmas concerning uniqueness and comparison.

Lemma 2.1 Suppose that ui E H2+'(8) , i = 1,2 are solutions of

A U + u [ ~ ( x ) - PU) = 0, X E 9

where I ( x ) is continuous in 5, p is apositive constant, and ui(x) > 0 in 9, = 0 on 89, for each i = 1,2. Then ui(x) ?= u2(x) for allx E 3. Proof . See [4], p. 17.

Lemma 2.2 Let Ij(x), i = 1 ,2 be continuous, Il (x) 2 I Z ( X ) > A1 for aN x E 3. Suppose that ui E H2+'(g) , i = 1,2 are respectively solutions of

AU + u [ l i ( ~ ) - P U ] = 0, X E 9

with u1 > 0 , u2 > 0 in 9, u1 = u2 = 0 on 89. (Herep is apositive constant.) Then u1 (x) 2 U ~ ( X ) for all x E 9.

Proof . The function w = ul is an upper solution for the problem A w + w[12(x) - pw] = Oin 9, w = Oon69, because Aul + ul[12 -pu l l = ul[ -I l

+ pul + I2 - pull < 0 in 3. Furthermore w = 60, is a lower solution for this same problem, for sufficiently small 6 > 0, because A (60) + 6w [ I 2 - p80] =

214 A. Leung

60 [ - A1 + I , - p60] 2 0 in 3 for small 6 > 0. By [8], there exists a solution w = ii2 to this problem, with 60 Q iiz < u1 in 5. Consequently U 2 > 0 in 9, and by Lemma 2.1 C2 = u2 in 8. This proves the lemma.

Remark 2.1 The existence of a solution of (Pl) with components which are positive in 9 can be established by the following procedures: For each u, 0 Q u

< - e + f - , hypothesis (H) implies that u = 6 o ( x ) is a lower solution of

the first equation in (Pl), for 6 > 0 small; and u = a/b is an upper solution. For each u, 0 < u < a/b, u = 60(x) is a lower solution of the second equation in

(Pl) for 6 > 0 small; and u = - e + f - is an upper solution. By [2] or [9], 9

b' one has a solution, (u, u) , with 60 (x) < u (x ) < - 60 (x) Q u ( x ) < - e + f - , all x E 3.

We now construct monotone sequences closing in to our solution. First, let ul E I f 2 + ' ( @ , be the unique strictly positive function in 9, satisfying A ul + u1 (a - bul) = 0 in 9, ul = 0 on 69. Note that such solution exists because 60 (x ) is a lower solution for small enough positive 6, and large positive constant functions are upper solutions; it is unique by Lemma 2.1. Similarly, let u1 E H Z C ' ( 8 ) , ul > 0 in 9 b e the unique such solution of A u1 + u1 (e + ful - gul) = 0 in 9, u1 = 0 on 69.

Lemma 2.3 u1 (x) < c-l (a - Al)for all x E 5. Proof. Since all constant functions larger than a/b are upper solutions of Au, + u1 (a - bul) = 0 in 9, u1 = 0 on 69, we have u1 ( x ) Q a/b for allx E 9, and

e + ful (x) < e + - for all x E 9. Consequently, constant functions w(x) .I

K 2 (e + 5) are upper solutions for A w + w (e + ful - gw) = 0 in 9,

w = 0 on 69. Serrin's sweeping principle implies that u1 (x) < - e + - On the other hand (H) implies that (1 - 5) a > Il + -, hence a > A1

+ - e + - a which clearly proves the lemma.

With the aid of the following lemma, we now define uir ui to be strictly

9 l ( i )

g 9 U 9

f a b

9

ce 9 ( ?)* 9

9 c ( i) positive functions in 9, i = 2,3, . . . inductively as follows:

Aui + ui(e + f u i - gui ) = 0 (2.1)

ui = ui = 0 on 69, ui, ui E H2+'($). Lemma 2.4 For each i = 1,2,. . ., ui > 0, ui > 0 in 9, and ui Q ul, ui < u1 in 0. They are uniquely defined in H2+'($).

A u ~ + U ; ( U - bui - c U j - 1 ) = 0 in 9


Proof . The lemma is clearly true for i = 1. Assume that it is true for i = j - 1. We have a - CUj-1 2 a - cul , which is > A 1 for x E $by Lemma 2.3. Therefore, comparing with lower solution 60 for small 6 > 0, we see that uj > 0 in 9% Comparing u, with ul by means of Lemma 2.2, we conclude uj < ul in 8. Again comparing with the lower solution 60 (small 6 > 0), utilizing the second equation in (2.1) with i = j , we see that uj > 0 in 9. Since uj Q ul , we compare the two equations which uj and u1 satisfy by means of Lemma 2.2, and conclude that uj < u1 for x E 5. This proves Lemma 2.4.

We will next observe some order relations among the sequences ui, ui, i = 1,2, ... Lemma 2.5 For each nonnegative integer n, the followirig are true:

(2*2) U 2 n + 2 < U 2 n + 4 < u 2 n + 3 < u 2 n + l , 0 2 n + 2 < u2n+4 < 0 2 n + 3 < 0 2 n + l

for all x E G. P r o o f . We first consider the case when n = 0. Using the equations which uI , u3 satisfy and utilizing Lemma 2.2, we conclude u3 Q u1 in 8. This in turn, using the same method, implies that u3 < uI in 3. Again, this implies that u2 Q u4 and then u2 < u4 in 8. To establish comparison between u3 and u4 or u3 and u4, we keep applying Lemma 2.2 to first assert u2 Q u1 in 8, then u2 Q u l , then u2 5 u3, then 02 Q u 3 , then u4 Q u3 and finally u4 Q u3 (all inequalities being true in 9). Ineqal- ities (2.2) are thus true for n = 0. Assume.that (2.2) are true for n = j - 1, using the above method and u2j < u2j+2 we deduce the following in order: ~ 2 , + 3 < deduce the following in order: u 2 j + 2 < uzj+2 5 ~ 2 , + 3 , u Z j + 2 Q 0 2 j + 3 , L(2,+4 < ~ 2 , + 3 , ~ 2 , + 4 < ~ 2 , + 3 . (All inequalities are true in 9.) This proves (2.2) for n = j .

Lemma 2.5 clearly implies that 0 < u2 < u4 5 2 4 . . . < u5 < u3 < u l , a n d O Q u 2 Q u 4 < V6 ... ~ u S ~ u 3 Q u l f o r a l l x ~ 9 . ~ e f i n e u * = l i m ~ 2 , , + ~ ,

u , = lim uZn, u* = lim u 2 n + l , and u, = lim uZn. The functions u , , u *, u, , u* are in H2+'(G) and (wl , w2, w 3 , w4) = (u, , u* , u, , u ' ) is a classical solution for the problem:

A w I + w l ( a - b w l - c w 4 ) = O , Aw3+ ~ 3 ( e + f w l - g ~ 3 ) = O

Aw2 + w2(a - bw2 - cw3) = 0 , Aw4 + w4(e + f w 2 - gw4) = 0 (2.3)

wi = 0 on 69 , i = 1,. . .4 . (See [8], p. 26, for the assertion on the regularity of w,) . Note that w&) > 0 for all x E 9.

We next investigate some relationship between problem (Pl) and problem (2.3).

Theorem 2.1 u, e u y in 5if f u, = u* in 4. P r o o f . Suppose u, = u* in 9, then u, and u* will be classical solutions to the - same boundary values problem. By Lemma 2.1, we conclude the u, = u* in 9. Similarly, the converse is also true.

u2j+l, u 2 j + 3 < U 2 j + l r u2j+2 < u2j+4, u 2 j + 2 < u 2 j + 4 . Then using u2j+2 < U Z j + l we

n- m

n-w n+w n- w

in 9

276 A. Leung

Theorem 2.2 Suppose that u, = u * in 3 (or u, = u* in 8), then the boundary ualue problem (Pi) has a unique nontrivial nonnegative solution in 3. More precisely, any solution (u, u ) with the property that u 2 0, u 2 0, both +O in 9 will satisfy (u, u ) = (u , , u * ) = (u* , u * ) in a’. Furthermore, u > 0 , u > 0 in 9.

Proof. Note that the existence of solution (u, u ) to (Pi) with nonnegative components in 9 had already been established in Remark 2.1, even without the assumption that u, = u * in 4. It remains to prove that under this assumption we must have (u, v ) = (u*, u * ) . Since u 2 0 in 9, using the first equation in (P1) and large constants as upper solutions, we conclude u Q a / b in a’. Subsequently,

using the second equation in (Pl) we can conclude that u < - e + f - in a’.

Hypothesis (H) therefore implies that a - cv > Il in 3. We next show that u > 0 in 9. Suppose not, and u (2) = 0 for .?E 9. Then the first equation in (Pl) implies that Au < 0 in a neighborhood of 9, and the maximum principle implies u s 0 in this neighborhood. This leads to the conclusion that u HI 0 in 5, contradicting our assumption. Therefore u > 0 in 9. Now, we can use Lemma 2.2 to conclude that u 6 u1 in 3 similarly 0 < u < ul in 9. Again, applying Lemma 2.2, we have u2 < u and v2 Q u in 8. Repeated applications of Lemma 2.2 will, by induction, show that u2,,+2 < u < u2n+ , uzn+2 G u G uZn+ x E a’, for each nonnegative integer n. Clearly, if u, 3 u* (or u* 3 u*) , then (u, u ) = (u*, u * ) .

The following theorem has also been. proved above.

9 Y :)

Theorem 2.3 Any solution (u, u) of the boundary ualueproblem (Pl), with both u and v 2 0, d0 in a’, must satisfy:

u* < u < u’, v* Q u < u* , for all x E a’. (Hence, u > 0, u > o in 9.1

Remark 2.2 Note that if we set u = 0 (or u = 0) in 6 and solve for v > 0 (or u > 0) in 9 in the second (or first) equation in (Pl), we obtain a nonnegative solution (u, u) . This is “trivial” in the sense that one component is =O.

The following partial converse of Theorem 2.2 is true.

Theorem 2.4 Suppose that the boundary ualue problem (2.3) has at most one solution with the property that all its components arepositiue in 9, then u, = u* and v , = v * i n a’. P r o 0 f. We have seen that (w1, w2, w3, w4) = (u*, u *, u,, u * ) is a solution of (2.3) with property as stated above. Moreover ( w l , w2, w 3 , w4) = (u *, u,, u *, VJ

is also such a solution. Hence, uniqueness implies that u, i~ u *, u , jii u* in 9.

3 Generalization to Systems of Order Higher than Two

We briefly outline a case where our monotone scheme can be generalized to the study of more than two interacting populations. We will only illustrate a


readily obtainable case, since the most general case is too lengthy for our present purpose. Consider the following system of two predators and one prey:

Air + ir(a - bir - ~ 1 6 - ~ 2 6 ) = 0

(P2) Aij + O(e + f i r - g10 - g26) = 0

A s + 9(P +fi - 918 - 5 2 6 ) = 0

f o r x e 9 , a n d C = O = 6=Oond9.Here,a,b,cl,c~,e,f,g~,g~,i?,~~~,and~~ are all positive constants. To insure the existence of positive solutions, we make the following assumptions:

(HI) 1 - (cifY(gib) - (~2f3/(52b) > 0 (H2) a > [Al + cle/gl + c26/G2][1 - (clf)/(gl b ) - (c&/(&b)] -' (H3) e > .II + g25F1 P +ye- ( ;)

Note that all these four hypotheses will be satisfied when a, e, 0 are large and that intraspecies interactions are large compared with interspecies interactions (i. e. gl and i 2 are large compared with il and g2 respectively, or with c1 and c2 respectively).

Let ir, , 01, 9, be functions in H2+'(G), which are strictly positive in 9, satisfying the following conditions:

(3.1) Air1 + ir,(a - b i t ) = 0 in 9, ill = 0 on d 9 ;

(3.2) AO1 + Bl(e + firl - glijl) = Oin 9, 61 = 0 on 8 9 ;

(3.3) A+, + ~ ~ ( i ? + firl - i2al) = 0 in 9, G~ = o on 69.

As in the argument immediately before Lemma 2.3 (also using assumptions (Hl) to (H4), the functions irl , t i l , G1 are uniquely defined and >O in 9.

Lemma 3.1 el (x) < 9;' (e + f f ) , $1 G ii' (0 + ff), and c181 (x) +

c2 $1 (x) < a - A ~ , for all x E 4. Proof . As in Lemma 2.3, we have ir(x) < a/b for all X E 4, and using (3.2) and

. Similarly, using

. The first two inequalities of this lemma

Serrin's sweeping principle we obtain O1 (x) < g;' (3.3), we obtain Ql < 5y1

and hypothesis (H2) imply the third inequality.

278 A. Leung

With the aid of the following lemma, we now define di, o^i, qi to be

Afi, + f i j (a - bfii - clSi-l - c2fiie1) = 0

ASi + Si(e + f f i i - g 1 S i - g2Gi-l) = 0

strictly positive functions in 4, i = 2,3 , . . . inductively as follows:

(3.4)

(3.5) in 9

(3.6) AGj + Gj(Z + ftij - jlSj-1 - &Gj) = 0

f i i = Si = $, = o on 84, ti,, bi, G~ E H ~ + ' ( G ) . Lemma3.2 Foreachi=1,2 ,..., f i i > O , S i > O , $,>Oin 9 ,andt i iQf i l ,BiQS1, Gi Q $1 in 3. they are uniquely defined in H2+'($) . Proof . Assume that the lemma is true for i = j - 1. We have a - c1 S j - l - C 2 Q j - 1 B a - Clu^l - ~ $ 1 > 11 for x E 3, by Lemma 3.1. As in Lemma 2.4, we derive fij > 0 in 9 a n d fi, Q til in 3. Next, note that e + f f i j - g2Gj-, 2 e - , \

g2Gl 2 e - g2g2 Z. + f - > A1 for x E 3, by Lemma 3.1 and (H3). As in the --T -3 final part of lemma 2.4, we derive that S j > 0 in 9and C j Q S1 in 5. Similarly, we prove analogous properties for + j .

Lemma 3.3 For each non-negative integer n, the following are true:

(3.7) c2n+2 < f i2n+4 < a2n+3 < f i2n+l .

(3.8)

(3-9)

82n+2 < ;2n+4 < ;2n+3 Q ;2n+l

*2n+2 < +2n+4 < *2n+3 < $2n+l

for all x E 4. Proof . We keep applying Lemma 2.2. First show ti2 ,< til, then fi2 Q Sl and fi2 < fi1. Then we establish the following in order: fi2 Q fi3 Q ti,, S2 < S3 ,< S,, G2 Q G3 Q G l , fi2 Q C4 Q fi3, B2 Q S4 < S3, G2 Q G4 Q G3. We thus have (3.7) to (3.9) for n = 0. Assume that the lemma is true for n = j , we then establish the following inequalities in order: &+4 Q &,+, Q i i 2 , + 3 , & j + 4 Q 8 2 j + 5 Q S 2 j + 3 , $2 j+4 6 i;'Zic5 Q

This proves that (3.7) to (3.9) are true for n = j + 1, and thus the lemma. %j+3, f i2 j+4 Q f i 2 j + 6 Q G 2 j + 5 , 6 2 j + 4 < 6 2 j + 6 < a 2 j + S 9 G 2 j + 4 < $ 2 j + 6 < 6 2 j + 5 .

We now have inequalities 0 Q ti2 Q ti4 Q t i 6 . . . Q 2, Q ti3 Q C l , 0 ,< S2 < 64 < 8 6 . m . < 6, < 53 < 81, 0 < $2 < G 4 < $6 ... $5 Q $3 6 $1. Let fi* = lirn fi2n+l, ti, = lim 02,, S* = lim S2n+l, d, = lim S2,,, $* = lim

G* = lim G2,,. Repeated applications of Lemma 2.2, as in Theorem 2.3, we can

prove that

Theorem 3.1 Any solutions (ti, 6, G ) of the boundary value problem (P2), wirh all ii, 5, G 2 0, 60 in G, must sathfy:

n-- n - 0 n-Ca n - 0 n - 0

n-+m

Monotone Schemes for Semihear Elliptic Systems Related to Ecology 279

a* < ti < a*, 8 , < s < s*, 9* < 9 < 9* for aii x E 3. Also, in case ti, = a*, 5, = s*, 9* = a*, then (ti, 6, 9) = (a*, 9*, a*).

Further investigations are certainly possible, but will not be pursued here.

4 Applications to Uniqueness and Stability

Returning to problem (Pl), we will see that under conditions additional to (H), uniqueness will result. Let a, b, e, g, C, F be fixed positive constants,

a > A1, e > A1 satisfying: CF < gb, a > gb(gb - CF) -' Al + - . Observe

that for all positive constants c, f with c < C, f < F, hypotheses (H) will be satisfied (with fixed a, b, e, 9). Let a 0, 3; t E H2+'(g) be strictly positive functions in 9, which are solutions to the following:

( 3 A O + O(a - b O ) = 0 in 9, O = O o n 6 9

A P + P e + F a - g P = 0 in 9, P = O o n 6 9

A D + + ( a - b 0 - C t ) = O i n 9 , 0 = O o n 6 9

A V + t ( e - g t ) = o in 9, t= O o n 6 9 .

( b ) (4.1)

a b (Note that such 0, c t exist because a, e, e + F - are > A l , and 0, t a r e

260 > 0 in 9, for sufficiently small 6 > 0. We can also prove as in section 2

, hencea - CP(x) 2 a -

all x E 3, by assumptions on C, F. Consequently 0 2 60 > 0 in 9 fo r 6 > 0 sufficiently small.) By Lemma 2.2, we have 0 < 60 < 0 < #, 0 < 60 < ,< P for all x E 9,6 > 0 sufficiently small. Since the outward normal derivatives of o are negative on the boundary, there must exist a constant K > 0 such that

(4.2) O < K 0 8 P < K t , O < K E V < K 0

for all x E .G. Now let c, f be arbitary numbers satisfying 0 < c < C, 0 < f < F (with

a, b, e, g fixed). For such choices, let ui, ui, i = 1,2, . . . be defined as in section 2. We have ul = 0, u1 < 0 < u2, t < u2 for x E 6, by Lemma 2.2. Moreover,

280 A. Lcung

for all x E 3. (Note that if c, f a re reduced, such inequalities will be unchanged, and 0, 0, P are unchanged.) For i 2 1, we have:


Then, we use (4.6), (4.7) and (4.2), (4.3) again to obtain:

Combining (4.8), (4.9) and using (4.2), (4.3) once more, we obtain:

for each integer i 2 1.

such that By means of (4.10), we conclude that if c, f with 0 < c < C, 0 < f < Fare

1 K

(4.11) cZ f 2 < 8 b 2 g 2 ,

then lim I (u2i+ - uzi+2) u2i+l uzi+2 d x = 0. By dominated convergence theorem,

and lim u2i+1 = u* > Oin 9, lim u2;+2 = u, > 0 in 9, bm (UZ;+~ - uZi+2) = u*

- u , 2 0 in 9, we conclude u* = u, for all x E 9. (Note that (4.2). (4.3) imply that K is unchanged by reducing c and J ) Applying Theorem 2.3, we therefore arrive at the following:

Theorem 4.1 Let hypothesis (H) be satisfied for boundary value problem (Pl). The solution (u, u ) of (Pi) with both u and v 2 0, +O in 9 b unique, prouided that (4.1 1) is satkfied.

To apply the monotone scheme in section 2 to a stability problem, we consider (u (x, t ) , v (x, t ) ) satisfying:

i-w 9

i- o 1-W 1-W

- A u + u(a - bu - cu) - - - A V + u(e +fu - go) au (4.12) - - at Bt

for (x, t ) E 9 x (0, m), where 9 and a, b, c, e, f, g satisfy all the conditions described in section 1. Let ui(x), ui(x), i = 1,2,. . . be as defined in section 2. For T > 0, let gT = 9 x (0, T). H2+'*(Z+o/2(8$) denotes the Banach space of all real-valued functions w having all derivatives of the form D"D:w (a is a multi- index, r 2 0 is an integer, D, = W a t ) with 2r + la1 Q 2 continuous on 6Tand having finite norm I w l r ' (as described in [lo]).

Theorem 4.2 Let i be an arbitrarypositiue integer. Suppose that (u (x, t ) , u (x, t ) ) is a solution of (4.12) in H2+L(2+o/2(gT), for each T > 0, satisfying:

282 A. L e u

for x E g, (4.14) u Z ~ ( X ) = u (x, t ) = ~ 2 i - l (x) = 0, u ~ ; ( x ) = U(X, t ) = ~ 2 i - l (x) = 0

for (x, t ) E 8 9 x [0, 03). Then (u (x, t ) , u (x, 1)) will satiyfy:

(4.15) ~ 2 j ( x ) < u(X, t ) 6 UZi-l(x) uzi(X) 6 v ( X , t ) < u ~ ~ - I ( x )

for ail (x, t ) E 3 x [o, 03). Proof . Observe that u2i, u ~ , - ~ , u2i, u2i-1 satisfy the following inequalities:

Av2; + uzi(e + f u Z i - gozi) - - au2i = 0 2 0, at

for all (x, t ) E 8 x [0, m) (here denote uo =I 0). These are the same as the four differential inequalities in (2.3) of [6], with ul, wl, u2, ~2respectively replaced by uzi, uti-1, UZ,, u2i-1, - r replaced by e, f 1 (p, q) = - bp - cq, fib, q ) = ep - gq. Note that f l , f z here will satisfy (1.3) in [6]. The fact that - r c 0 is replaced by e > 0 will in no way affect the proof of Lemma 2.4 in [6]. Applying Lemma 2.4 of [6], with (ul (x, t), u2(x, t)) replaced by (u(x, t ) , u(x, t)), we obtain inequalities (4.15).

Theorem 4.2 gives a family of spatially dependent invariant regions closing in to the set {(x, u, u ) Ix E g, u,(x) 6 u < u*(x), u,(x) < u < v* (x)}. In case (4.11) is satisfied, the set becomes {(x, u,(x), U,(X)(XE 9, since u, = u*, u , = u*. For each x E 9, let lf = {(x, U, 0) I u ~ ; ( x ) < u < ~ 2 i - 1 (XI, 02j (X) 6 0 < u2i-l (x)}, for each positive integer i. Clearly, Lemma 2.5 implies that 5; 1: for each X E 9. The following remark will show that the set inclusion is proper.

Remark 4.1 In Lemma 2.2, suppose that we have the additional assumption that I , (x) # I2(x), x E 9. Then u1 (x) > U Z ( X ) for all x E 9. Consequently, our construction of monotone scheme will imply that the following inequalities are strict:

(4.16)

for each x E 9. (Observe here that u1, u2 in (4.16) play different roles as those in the beginning of the remark.)

0 c u2(x) c uq(x) < . * - c us(x) c u3(x) c Ul(X)

0 < uz(x) < uq(x) c * - . c +(x) c u3(x) c q ( x )


To prove the assertion of the above remark, we note that w = u1 is an upper solution for: A w + w[l2(x) - pw] = 0 in 9, w = 0 on 69. We proved in Lemma 2.2 that 60 < u2 < u1 for sufficiently small 6 > 0. We conclude the existence of such u2 by starting to iterate from the upper solution w = u1 (a procedure described in p. 25, [SJ). The property that the upper solution u1 satisfies the boundary condition exactly at 6 9 will imply that the next iterate is strictly less that the upper solution u1 inside 9(because of maximum principle), unless Au, + u1[12(x) - pull = 0 in 9. However Aul + u1 [12(x) - pull = ul [ - 1, (x) + 1, (x)] f 0 in 9. Consequently, if 1, (x) Pa 12 (x) in 9, we must have u2(x) < u, (x) in 9, in Lemma 2.2.

In view of (4.16) we observe that for each x E 9, If+l C interior of Iffor each integer i 2 1, and

{(x, u, u ) I u * (x) < u < u * (x), u * (x) < u < u * (x)) c * * * c ri" c . . * 1; c 1:.

In view of Theorems 4.2 and 4.1, one might say that the positive equilibrium solution of (4.12) with u = u = 0 on 69 x [0, m) is stable, when (4.11) is satisfied.

5 Examples and Remarks

The Volterra-Lotka type of interactions in ecological problems have been widely studied lately, see e.g. Il l ) , [12], [15]. The study of equation (Pl) considers the diffusive effect on this type of interactions. Such effects have received much attention currently see e.g. 131, [6], [lo], [13], [14]. The three species interactions considered in (P2) of section 3 correspond to case 18 in [12].

As an example for (Pl), we note that the first condition in (H) is easily satisfied, and the second condition in (H) is readily satisfied when the parameter a is large. When n = 1 , 9 = (0, n), an example will be:

+ v ( 3 + u - 2 u ) = 0 for XE (0, x ) d2u dx2

+ ~ ( 5 - 2 ~ - ~ ) = 0 - d2u (5.1) - dx2

with u(0) = u(0 ) = u(x) = u(n) = 0. (Here, R1 = 1.) Consequently, the sequences constructed in section 2 will be monotone. All results in section 2 are applicable.

As an illustration for an application of section 4, we consider

d2u d2u d x dx2

+u(3+0.012u-2u)=O, o<x<n (5.2) 7 + ~ ( 5 - 2 ~ - 0 . 0 4 4 ~ ) = 0 -

with the same boundary condition as (5.1). Defme 0, 0, by means of formulas in ( 4 4 , with a = 5, b = 2, C = 1 , e = 3, F = 1 , g = 2. (Note that such choice of parameters satisfies condition (H) as seen in (5.1), and observe that C = F = 1 .) By using upper and lower solutions method, we obtain 0 < o(x) < 5/2, 0 < P(x) < 11/4 for all x in [0, x ] . Using the fust equation in (4.1), we obtain I d 2 o/dx2 I < max I u (5 - 224) I = 2518; and using the bounds of 0 and its

O( u (J/2 second derivative, we obtain

284 A. Leung

for all x in [O, x]. (See for example [16], p. 139, for such estimates.) Similarly, we deduce that Id2 v/dx21 < 3.79 and Id v/dxl < 7.71 for 0 < x < R. Since 5 - P 2 2.25, the third equation in (4.1) gives (0.625) sinx Q o ( x ) Q 2.5; similarly sinx Q P(x) < 1.5, for all 0 < x < x. For 0 Q x Q 0.384, we have 0 < o ( x ) < 6.51 x. The fact that (2.5/sin 0.384) < 6.68 implies that 6.68 sinx 2 o ( x ) for all 0 < x < x (since 6.68 sin (0.384) > 6.51 (0.384), and o ( x ) Q 2.5). We have the inequality:

(5.3)

since sinx < v(x). Similarly, we deduce that

(5.4) P(x) < 12.65 o(x) for 0 Q x Q x . Let us refine inequalities (4.2) to

o(x) < 6.68 P(x) for 0 Q x < x ,

O < K * R P Q K #

for some positive constants Kl , K2. The factors K2 in (4.8) and (4.9) can all be replaced by K:. Similarly, the factors K8 in (4.10) and (4.1 1 ) can be replaced by Kf * Ki. (Again, note that K1, K2 are unchanged by reducing C, F to c, f.) The improved version of hypothesis (4.11), in this case, becomes

b2 g2 1

(6.68)4 (12.65)4 (5 .5 ) c2f2 <

with b = g = 2. (5.5) is clearly satisfied by choosing c = 0.044 and f = 0.012 in equations (5.2). We can therefore apply Theorem 4.1 (with (4.11) replaced by (5.5)) to conclude the uniqueness of positive solution for problem (5.2). We also conclude that the solution is stable, as explained in Remark 4.1. We note that our estimates leading to inequalities (5.3) and (5.4) are very crude. With more careful calculations, one should be able to improve (5.3), (5.4) and thus (5.5). One should therefore be able to apply Theorem 4.1 to conclude uniqueness for larger choices c andf. Finally, we can apply Theorem 4.1 without first improving (4.11) to (5.5). In such case, we might need smaller choices of c and f to conclude uniqueness. However, condition (4.11) is always satisfied provided c, f are chosen sufficiently small.

Acknowledgment

The author wishes to thank Professor A. C. Lazer for calling his attension to reference [4] and valuable discussions on the subject. The author also thanks the editor and referee for their encouragement in writing section 4.


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Resources. New York: Wiley 1976

33 (1977) 673 -686

Anthony Leung Department of Mathematical Sciences University of Cincinnati Cincinnati, Ohio 45221 (Received June 3, 1981)

monotone schemes for semilinear elliptic systems related to ecology

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