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Existence of global coexistence state for periodic competition diffusion systems
Nonlinear Analysis, Theory, Printed in Great Britain.
Methods
& Applicolions,
Vol.
19. No. 4. pp. 335-344,
1992 0
EXISTENCE PERIODIC
OF GLOBAL COEXISTENCE COMPETITION DIFFUSION
0362-546X/92 $5.00 + .@I 1992 Pergamon Press Ltd
STATE FOR SYSTEMS-f
ANTONIO TINEO Universidad de Los Andes, Facultad de Ciencias, Departamento de Matematicas, Merida 5101, Venezuela (Received 1 October 1990; received in Key words andphrases:
revisedform25 Juty 1991; receivedfor publication 8 November 1991)
Periodic, reaction-diffusion,
global stability, asymptotic behaviour, extinction.
0. INTRODUCTION IN THIS
paper we consider the following system of parabolic equations: U, = P(U) + u[a(t, x) - b@, x)u - c(t, x)u] u, = Q(u) + v[d(t, x) - e(t, x)u - f(t, x)v]
(0.1)
where a , . . . , f: R x $2T, R are positive continuous functions, which are periodic in the time t with period T > 0; Sz is a bounded domain f R”whose boundary 6X2belongs to C2 and P, Q are uniformly elliptic operators in R x fi, whose coefficients are T-periodic in the first variable. When P, Q have the form k(t, x) A(u); where A denotes the Laplacian with respect to the space variables; the system above models the competition between two species, with migration from regions of high population density to regions of low population density. For details see Pao [4] and the references therein. When a , . . . , f and %2 are sufficiently smooth, we shall study the existence and uniqueness of a positive solution (u, u) (U > 0, u > 0) of (0.1) such that v(t + T,x) = u(t,x)
u(t + T, x) = u(t, x), adav
= adav
= 0
in IRx an
(0.2) (0.3)
where a/av denotes differentiation in the direction of the outer normal to X2. We also study the asymptotic behaviour of positive solutions (u, u) of (0. I), which satisfy the following initial boundary value problem: adav
= adav
MO, x) = d(x),
= 0
in (0, 00) x asz
u(O, x) = w(x)
in !2
(0.4) (0.5)
for sufficiently smooth nonnegative functions 4, I,U:d --f R such that a+/av = av/av = 0, in an. To be precise, let us define from now on: B = b/a, C = c/a, E = e/d, F = f/d, and for all bounded functions g: X + R (some nonempty set X); g, = inf(g(x): x E X) and g, = sup(g). In the “stable” case: and FL > CM, (0.6) BL 7 EM j-This work was sponsored by the C.D.C.H.T.
of the Universidad de Los Andes, Venezuela. 335
336
A. TWEO
we prove the existence of a positive solution to (O.l)-(0.3) and we obtain a priori bounds for the positive solutions to this problem, whose components belong to C**‘(IR x a) n C’*‘([R x ai). Actually, our results generalize theorems 1.1 and 4.1 of Ahmad and Lazer [ 11. Further, if (0.6) and C,u&Xu
- C,)(B,
- EL) <
&&FL
-
Cd@,
- &I
(0.7)
hold, then the problem (O.l)-(0.3) has exactly one positive solution. Moreover, this solution is globally asymptotically stable. It is a partial answer to a question in [l]. Remark.
If B, C, E, Fare constant, then (0.6) implies (0.7).
Remark.
If 2 maxlB,C,,
FMEM] I BLFL, then (0.6)-(0.7) holds.
Proof. Notice first that BL L 2E, and so, BL - EM 2 (l/2)8,. Similarly, FL 2 2C, and FL - CM L (1/2)F,. Thus, (0.6) holds and B,F,(F, - C,)(B, - EM) 2 (l/4)(BLFL)* 2 B,C,E,F, > C,E,(F, - C,)(B, - EL). So the proof is complete.
When EL > BM (resp. C, > FM) we study “the domain of attraction of the trivial solution (Uo, 0) (resp. (0, V,))“, where U, (resp. V,) is the unique T-periodic solution to the equation U, = P(u) + u[a - bu] (resp. u, = Q(v) + u[d - fv]), which satisfies the Neumann boundary condition in IRx &X We used these results to improve a theorem in [I], about the extinction of one species. To complete our paper, we exhibit an iterative scheme which allows us to find a positive solution to (O.l)-(0.3), when a > cVo and d > eU,. We shall see that these conditions are implied by (0.6). 1. A PRIORI ESTIMATES In the following, P, Q, R denote elliptic operators of the form
(1.1) where aij, bi: IRx fi -+ R are continuous functions which are T-periodic in the time t and aij = aji. We also assume that there exists E > 0 such that n n C aij(f9 x>titj 2 E C ti’ i,j=l
for all
i=l
in this paper are assumed to be continuous
PROPOSITION1.1. Let w E C* := C*,’ (R x a) fl C”‘(IR x ai) be a nonnegative solution to in m x an. w, = R(w) + gw, aw/av = 0
and
T-periodic
If w # 0 (resp. wL > 0), then there exists (tr, x,) in IRx fi such that w(t,, x,) = w, (resp. w(t,, xl) = wL) and g(tr ,x1) L 0 (resp. g(t,, x1) I 0). Proof.
It follows from the same arguments in [l, lemma 1.21.
Global coexistence state
337
PROPOSITION1.2. Let (u, u) be a nonnegative solution to (O.l)-(0.3) such that u, u E C*. (a) If uL > 0 and uM > 0, then FL - CM I (B,F,
- CMEL)uL
(1.2)
BM - EL 2 (BMFL - C,E,)v,.
(1.3)
(b) If uM > 0 and uL > 0, then FM - CL I (B,F, Proof.
and
- C,E,)u,
We only prove (a). From proposition
BL - EM I (B,F,
- CL E&v,.
1.1, there exist (t, ,x1), (tz, x2) in R x fi such
that 00, , x1) 5 w, 9x1 1% + co, 9x,Mt, w,
3$1 1 4t, 3XMf2
From here, 1 i BMuL + C,u, If B,F,
- C,E,
and 1 L E,u,
# 0 and B,F,
- C,E,
9x1)
, x2) + _m* 3x2h.f. + FL uM; and the proof follows easily.
# 0; we define
E, = (BMFL - C,E,)-‘(FL
- CM)
6, = (&FM - GE,)-‘(FM
- CL)
E, = (B,F,
- EM)
(1.4) - GE,)-‘(B,
6, = (BMFL - &EL)-‘(B,
- EL).
We have the following immediate corollaries to proposition
1.2.
COROLLARY 1.3. If (0.6) holds and (u, u) is a positive solution to (O.l)-(0.3), whose components belong to C*, then E, I u I 6, and E, s u I 6,. Remark.
Assume bL/aM > e,/d, (FM - GM&F,
and fL/dM > c,/a,.
Then, BL > EM and hence
- &CL,) 5 tfM/dL- c,/%M&f,/&
- EIMcJcIIM)
= (aMf,tt - c~&Y(& aMfM- Ed,&) 5 (aMfM- cLdMbLfM - CL%,& since FM d fM/dL and C, 2 CL/a,. [l, theorem 1.11.
From this, it is easy to prove that corollary 1.3 generalizes
COROLLARY 1.4. If B, C, E, Fare constant, then the constant (u, u) = (E,, 6,) (= (6,) E,)) is the unique positive solution to (O.l)-(0.3) in C*. COROLLARY 1.5. Assume FL 1 CM,
BMIEL,
(FL 94~) # (CM, EL)
(1.5)
and let (u, u) be a solution to (O.l)-(0.3) such that u > 0, u > 0 and u, u E C*. Then u = 0. We have a parallel result if: FM I CL, B, 1 EM and (FM, BL) f (CL, EIM).
A. TINEO
338
Proof. Suppose that vM > 0. From (1.2) we get, B,F, - C,E, > 0 and by (1.3), we obtain the reverse inequality. Thus, B,F, - C,E, = 0 and by (1.2)-(1.5), (FL, B,) = (CM, EL). This contradiction proves that v = 0. The other case is proved by the same arguments, and so the proof is complete. 2. UNIQUENESS
In this section, we prove that the conditions (0.6) and (0.7) imply the existence positive solution (u, v) to (O.l)-(0.3), such that U, v E C*. To this end, we need the following result. LEMMA
2.1. Let ((Y,p) be a T-periodic
solution
to the linear
CY~ = P(a) - ga + h/3,
P, =
adav = ap/av = 0 where g, h, k, I are positive
(T-periodic
continuous
problem:
Q(P) + ka - IP in IR x aa
functions)
. sup(k/l)
sup(h/g)
of at most one
defined
(2.1) (2.2)
in iR x 0 and
< 1.
(2.3)
If CY,p E C*, then (Y = p = 0. Proof.
Let us fix positive c 2 maxlaM,
constants
inf(k//)P,)
and
E < g, [l - sup(h/g)
sup(k/l)].
It is not hard to verify that (p(t), o(t)) := cexp(-et)(l, sup(k/l)) is a supersolution to (2.1) and (2.2) such that 01~ 5 p(O) and PM 5 o(O). Hence, cy 5 p and p I o(, since (2.1) is quasi-monotone. Assume now that CX(T,x) > 0 for some (r, x) E IR x fi. Then, for all integers n 2 1; c L 01(r + nT, x) exp(r + nT)
= cly(r, x) exp(r + nT)
+ co
asn+m
and this contradiction proves that (Y I 0. Analogously, /? 5 0. Repeating the proof with (--a, -p), we get, (Y 2 0, p L 0; and so the proof THEOREM
positive Proof.
2.2. If both
(0.6) and (0.7) hold, (24, v) E C* x C*.
solution Suppose
then
the problem
that (u, v), (U, V) E C* x C* are positive (Y = (u/U)
From an easy calculation
- 1,
(O.l)-(0.3)
solutions
is complete.
has at most
to (O.l)-(0.3)
one
and define
p = 1 - (v/V).
we get,
Olt = Po(cr) - bus
+ (a4 V/U)P,
P, =
Q,,(P) + (euU/Va
-fvP
where PO, Q. are elliptic operators which have the same principal part as P, Q respectively. Let E”, E,, 6,, 6, be defined by (1.4). Then, corollary 1.3 and (0.7) imply: sup(cV/bU) and the proof
sup(eU/fV) follows
= sup(CV/BU)
from lemma
sup(EU/FV)
2.1, since (Y,j3 satisfy
I C,B;‘~,E;~E,,.,F~~B,,E;~ (2.2).
< 1;
Globalcoexistence state
339
3.SOMEBASlCCOMPARISONRESULTS In the next section we assume that there exists CYE (0, 1) such that the coefficients of our equations belong to C 01,01”([0, T] x 0) and the coefficients aij(t, x), in (1. l), are differentiable with respect to the space variables and aa,/ax, E C uPn’2([0, T] x a), for all 1 I i,j, k I n. Moreover, we assume JKJ E Cm+*. From theorems 2.3 of [3] and 2.1 of [l] we get the following theorem.
3.1. Suppose that g, h > 0 on R x b. (a) If $I E P’*(fi) is nonnegative and a$/& = 0 in K!, then the problem:
THEOREM
W, = R(W) + w]g - hw],
at-dav
w(0, x) = d(x)
in (0, 00) X Q
= 0
(3.1)
in fi
(3.2)
has exactly a nonnegative solution w = w+,in CF := C 2+a,1+a’2((0, r] x d) n C([O, r] x Cl), for all r > 0. (b) The problem (3.1) has exactly one positive and T-periodic solution w, such that w * E C2+a*‘+u’2([0, r] x Cl) for all r > 0. Notice that, by proposition 1.1, we have inf(g/h)
I w, 5 sup(g/h).
(c) w,(t, x) - w,(t, x) -+ 0 as t --, co, uniformly
(3.3)
on fi; where wm and w, are as above.
In the following, I denotes the subset of C,‘*(a) consisting such that &$/av = 0 in aG. We identify the constant functions From [l, theorem 3.11, we get the following theorem. 3.2. Let ii, _v, 9, 6: [0, co) x fi + R be functions
THEOREM
of all nonnegative in I with [0,03).
functions
r#~
in
C*,‘((O, r] x Cl>n C’*O((O, r] x b) n C([O, r] x b),
for all r > 0,
such that ii 2 u L 0, b L u 2 0, iii, 2 P(U) + ii[a - bii - cg], aii/av
2 0 2 agav
_v, = Q(g) + _v[d - eti - fg] in (0, r] x an
and (u, 3) satisfies the reverse inequalities above. If 4, w E I; ~(0, x) I 4(x) 5 ii(0, x) and _v(O,x) I v(x) 4 U(O, x) in d; then the problem (0. l), (0.4), (0.5) = (0.1, 4, 5) has exactly one solution (u, u) E Cp x Cp for all r > 0. Moreover, g I u I ii and _v I v 5 V in [0, 00) x Q. 3.3. For each @I,I+YE I, the problem (0.1,4, 5) has exactly one solution (u, v) = urn,&), such that U, u E C,YY,for all r > 0. Moreover, if U, (resp. V+) is the unique solution (%,ti, to the equation U, = P(U) + U[a - bU] (resp. K = Q(V) + V[d - fV]) such that U(0, x) = cp(x) (resp. V(0, x) = w(x)) and aU/av = 0 (resp. aV/av = 0) in [0, M) x Sz, then u I U, and u I V, on (0, 00) x i2. COROLLARY
Proof.
It follows
from theorem
3.2, with (ii, _v) = (U4, 0) and (u, V) = (0, I$,).
340
A. TINEO
3.4. If (ul, v,),(u,, u2) are nonnegative solutions to (O.l), (0.4) such that ui, ui E C,? for all r > 0 (i = 1,2) and ur (0, x) 5 1.4~ (0, x) and ~(0, x) I vr (0, x) then, ur 5 u2 and v2 I v, in [0, co) x a.
COROLLARY
Proof. Let US define hi = ~~(0, X) and Ui = U,,, for i = 1,2; where U+ is as above. By [6, theorem 10.11, we get CJ, I U, in [0, 00) x Q, since 4r I &. By corollary 3.3, we have U, 5 U, in [0, co) x 52, and then (ii, _v):= (U,, 0), (g, 13):= (uI , v,), (4, I,V):= (z.+(0, *), v2(0, e)) satisfy the hypotheses in theorem 3.2. The proof now follows from this theorem.
COROLLARY
3.5. Let
(u, v) be a nonnegative solution to (O.l), (0.4) such that U, v E Cp, for all
r > 0, and ~(0, x) 5 u( T, x) and ~(0, x) 2 v( T, x) in Q. Then there exists a nonnegative solution (u* > v*) to (O.l)-(0.3) such that u*, v* belong to C 2+n*1+a’2([0, r] x Cl) for all r > 0; and
as t -+ to
(u(t, X) - u*(t, X)7v(t, X) - v*(t, X)) -+ (090) uniformly on b. Moreover,
(3.4)
U* 2 u and v, I v in [0, 00) x Q.
Proof. For each integer k 2 1, let (Uk, v,J be the solution ‘to (O.l), (0.4) defined by uk (t, x) = u(t + kT, x) and v,(t, x) = v(t + kT, x). By corollary 3.4; u I U, and v 2 vr in [0, 00) x Q, and then ur I u2 I ... and vr 2 v2 2 .... The proof now follows from the arguments in [ 1, theorem 3.21. Remark.
Let (u, v) be as above and suppose that there exists E > 0 such that u(t, x) r E in [0, 00) x Q. If v(t, x) + 0 as t -+ 03 then, v, = 0 and U, is the unique T-periodic positive solution to the equation u, = P(u) + u[a - bu] such that au,/& = 0 in R x dQ. 4,ANITERATlVESCHEME
Let us assume a > cV,,
d>eU,
(4.1)
where U,( V,) is the unique T-periodic positive solution to the equation U, = P(u) + u[a - bu] (v, = Q(V) + v[d - fv]) which satisfies aU,/av = 0 (a&/h = 0) in R x aQ. We shall prove that the following iterative scheme (4.2) vkf = uk(t
+
T, x)
=
Q(vk)
+
uk(t,
x),
vk W vk(t
auk /av = at&/& = 0 provides a positive solution to (O.l)-(0.3). To do this we need the following result.
euk +
- fvkl
T,x)
= vk(f,X)
in R x ai-2
(4.3) (4.4)
Global coexistence state
341
PROPOSITION4.1. Let g,, g, , h: IR x fi -+ IR be T-periodic continuous functions such that h > 0 and g, 5 g, . If Ui E C* is a T-periodic positive solution to the equation: u, = R(u) + u[gi - hu], such that aUi/av = 0 in iR x 80 then, U, 5 U,. Proof.
Let us write V = U/U,,
- 1. Then,
v, = R,(V)
- hU,V + (sr - g,)UJU,
(4.5)
for some second order elliptic operator which has the same principal part as R, and a V/av = 0 in R x an. Assume now that V, < 0 and define IV@,X) = V(t, x) - Vr.. Then W, = 0 and R,(W) - w - hU, W = hUV, + (go - g,)U,/U,, < 0. From [6, theorems 9.6 and 9.121, we get W = 0, and so I/ = V,. By (4.5), 0 I g, - g, = hU,VL < 0 and this contradiction shows that V 2 0. Thus the proof is complete. THEOREM4.2. If (4.1) holds, then problem (O.l)-(0.3) has positive solutions (ii, _v),(u, ii), both of whose components belong to C 2+a,1+a’2(lRx ai), such that: u I u 5 ii and _vI v I z7 for any positive solution (2.4,u) E C* X C* to (O.lHO.3). Proof. We shall prove the existence of two sequences (u,), (uk) of positive functions in C2+a*1+a’2(lRx G) which satisfy (4.2)-(4.4) and u1 2 u2 L ++e; V, I v2 I -.. I V,. Define u, = U,. Then, d - eu, > 0 and by theorem 3.1, the equation
u, = Q(u) + v[(d - eui) - fv] has exactly a positive T-periodic solution vr which satisfies the Neumann boundary condition. By proposition 4.1, u, I V0 and then a - cv, 2 a - cV, > 0. Consequently, the equation: u, = P(u) + ~[(a - cul) - bu] has exactly one positive T-periodic solution u2 which satisfies the Neumann boundary condition. Notice that, by proposition 4.1, u, 2 u2. In this way, we obtain sequences (u,), (u,J with the required properties. From somewhat well-known arguments (see [l, 31) we can prove that (z+., ok) converges to a solution (ii, _v)of (O.l)-(0.3). Notice that _vL u1 > 0. Let u, be the unique positive T-periodic solution to the equation u, = P(u) + u[(a - CV,) - bu] such that au,/av = 0 in R x aa. By proposition 4.1, we get u, r u,, since u, I V,. Thus, ii 2 u, > 0. Assume now that (u, u) is a positive solution to (O.l)-(0.3) such that u, u E C*. Since u 2 u0 = 0, then u, 1 u, and by proposition 4.1, ui I u. By induction, we have u, 1. u and u, I u for all integers n L 1; and so, ii 2 u and _vI u. To end the proof, it suffices to consider the following iterative scheme: ug = 0,
u,t = Q&J + u, id - eu,-, - _Ikl
Unt = P(u,)
+ u, [a - cu, - bu,].
So the proof is complete. Remark. Assume (0.6). From (3.3), we have V,, _ -= l/F’ and so, a - cV, = a(1 - CV,) 2 a,(1 - CM/F’) > 0. Analogously, d - eU, > 0, and thus (0.6) implies (4.1).
A. TJNEO
342
5. ON THE
DOMAIN
OF ATTRACTION
OF THE
Given a nonnegative solution (u,, u,) W(u,, u,) as the subset of I x I consisting
TRIVIAL
SOLUTIONS
to (O.l)-(0.3) we define (the of all points (4, w) such that
(u(r, x) - u*(t, x), u(t, x) - u*(t, x)) + (0,O)
(Cz’,,,O) AND
“stable
(0, I”)
manifold”)
as t + cc)
uniformly in ai; where (u, u) is the unique solution to (O.l), (0.4) and (0.5). It is clear that: (4,O) E W(U,, 0) and (0, I,U)E W(0, V,) for all 4, v/ E I; 4 f 0, I+Vf 0. 5.1. If (&, t+vO)E W(Og , 0) then, result holds for (0, 6).
PROPOSITION
A parallel Proof. w = vO). uniformly On the
(4, I,U)E W(U,, 0) for all 4 L do and I,UI wO.
Let (u, u) ((u,, uO)) be the solution to (O.l), (0.4) and (0.5) (with 4 = &, and From corollary 3.4 we know that u0 I u and u0 2 u and hence, u(t, x) + 0 as t + 03 in d. other hand, if U, is defined as in corollary 3.3, then, u 5 U, and U,#,(&X) - 060,x)
as t -+ to, uniformly
PROPOSITION
in 0. The proof
(4, V) belong Proof. B,4
EL L B,
5.2. Assume
if FL 2 CM and (A,,p(,)
= (~~,a,)
follows
+ 0
now from the following
inequalities:
and FL > CM if EL = BM and define (A,, po) = (0, l/C,) if F’ < CM. If 4, v/ E I and +r > A, and wM < p. then,
to w(Uo, 0).
Let Tbe the set consisting
of all points
(6, I,U)E R2 such that 4 > A,; 0 I I,V< ,u, and
+ C,v
I 1 I EL+ + F,y. If (@, I,V)in T then, (u, V) := ($, I,u), (ii, _v) := (B;‘,
0) satisfy the hypotheses in theorem 3.2 and hence 4 5 u(t, x) 5 l/B, and 0 5 u(t, x) 5 I,Uin [0, 00) x a; where (u, u) is the solution to (0.11, (0.4) and (0.5). In particular, u(T,x) 2 u(O,x) = 4 and u(T,x) I u(O,x) = cy. Let (u,, u,) be given by corollary 3.5. Then, U, 2 4 > 0 and by corollary 1.5, u, = 0 if FL I C,. Assume now that FL < CM. If uM > 0 then, by (1.2), A0 = E, 2 uL 2 4 and this contradiction proves that u, = 0. Consequently, U, = U. and hence, T c W( U, , 0). By proposition 5.1, (A0 , 00) x [0, po) c W(U,, 0) and the proof follows from this proposition.
To obtain
our results about the extinction
of one species, we need the following
proposition.
PROPOSITION 5.3. Let (u, u) be the solution to (O.l), (0.4) and (0.5); 4, v/ E I. If $ + 0 (I,Vf 0) then, u(t, x) > 0 (u(t, x) > 0) for all t > 0 and x E b. In particular, if (u, u) is nonnegative and T-periodic and u f 0 (u f 0) then u > 0 (u > 0).
Proof. We know that U, u are bounded functions on [0, 03) x d (see theorem 3.1 and corollary 3.3). So, there is a constant y > 0 such that y + Q - bu - cv > 0 in [0, 03) x d and thus, P(u) - u, - yu s 0 in [0, co) x b. The proof follows now from [6, theorems 9.6 and 9.121. See [l, proof of theorem 3.21.
Global coexistence state THEOREM 5.4.
343
If FL > CM and BM < EL then, (4, I,@E W(U,, 0) for all 9, w E I; 4 f 0.
Proof. Let 4, w be as above and let k’ = V, be defined as in corollary 3.3. We know that V(t,x) - V,(t,x) + 0 as t -+ 00, uniformly on fi and V, 5 Z$’ < CG’. Consequently, there exists an integer k 2 1 such that V(kT, x) < CG’ for all x E a. Let (u, v) be the solution to (O.l), (0.4) and (0.5) and define
u,(t,x)
= u(t + kT,x),
u1(t, x) = u(t + kT, x).
Then, (ui , II,) is a solution to (O.l), (0.4) and (0.5) such that u1 (0, x) > 0 (proposition 5.3) and vi (0, x) < C,&’ (corollary 3.3) in a. By proposition 5.2, we get (ui (t, x) - U,(t, x), vi (t, x)) + (0,O) as t -+ to uniformly on 0, and so the same holds for (u, u), since U, is T-periodic. The proof is thereby complete. From the change of variables (u, u) + (u, u) we obtain the following theorem. THEOREM
5.5.
If B, > EM and FM 5 CL, then (4, v) E W(0, V,) for all 4, v/ E I; y f 0.
PROPOSITION 5.6. If BL > E,,,,, then W(U,,, 0) = ((4, 0): 4 E I and 4 f 0). (A parallel result holds for (0, I$) if FL > CM.)
Proof. Let T be the set consisting of all pairs (4, r+~)E Rz such that 4 1 0, I,Y> 0 and EM4 + FMw I 1 I BL$ + CL I,Y;and fix (4, I+Y) in T. By the arguments in proposition 5.2, we
have that the solution (u, u) to (O.l), (0.4) and (0.5) is asymptotic, as t -+ 00; uniformly in d; to a nonnegative solution (u,, u,) to (O.l)-(0.3) such that u, > 0. So, T fl W(U,, 0) = $I, and the proof follows from the arguments in proposition 5.2. The results in this section Lotka-Volterra system.
are parallel
6. GLOBAL
to some results in [5], about
the ordinary
COEXISTENCE
From the arguments in Section 5 and theorem 4.1 of [l], we can prove the following theorem. If (0.6) holds and (ii, _v),(_u,V)are as in theorem 4.2, then for each solution (u, u) to (O.l), (0.4) and (0.5); 4, ry E I; 4 f 0, I+Y f 0; and for E > 0, there exists I, > 0 such that THEOREM 6.1.
u(t, x) - & I u(t, x) 5 ii(t, X) + E _v@,X) - 6 5 u(t, x) 5 a(t, x) + 6 if t > t, and x E Q. Remark. By corollary 3.4, we have that (4, w) E W(ii, _v) if 4 I ci(0, .) and y 2 ~(0, a). An analogous remark holds for (u, 5).
From theorems 2.2 and 6.1 we have the following theorem.
344
A. TINEO
THEOREM6.2. Assume (0.6) and (0.7). Then (O.l)-(0.3) has a unique positive solution (u,, uo). Moreover, W(U,, u,,) = ((4, t,~)E Z x I: $J f 0 and w f 0). COROLLARY 6.3. Assume BL > EM, FL > CM, inf(b/e) > sup(c/f ). If a, . . . , f depend only on t, then ii = u, v = B and ii, _vdo not depend on x.
Proof. In theorem 3.1, assume that g, h depend only on t and let w0 be the unique T-periodic and positive solution to: x’ = x[g - hx]. It is clear that w,, is a solution to (3.1) (with R = P) and so W, = w,,; where w* is given by theorem 3.1. So, w, does not depend on x. From this, we have that the same holds for the sequences (u,), (Q) in theorem 4.2, and then (ii, _v)depend only on t. A parallel result holds for (u, 0) and thus (ii, _v),(F, V)are T-periodic positive solutions to the following Lotka-Volterra system: U’ = u[a(t) - b(t)u
-
c(t)v]
u’ = u[d(t) - e(t)24 - f(t)u]. The proof follows from [2, theorem 2.11. Acknowledgement-The
author
thanks
the referee for useful comments.
The shorter
proof
of lemma 2.1 is due to him.
REFERENCES 1. AHMAD S. & LAZER A., Asymptotic behaviour of solutions of periodic competition diffusion systems, Nonlinear Analysis 13, 263-284 (1989). 2. ALVAREZ C. & TINEO A., Asymptotically stable solutions of Lotka-Volterra equations, Rad. Mat. 4, 309-319 (1988). 3. PAO C. V., Positive solutions of nonlinear boundary value problems of parabolic type, J. dvf. Eqns 22, 145-163 (1976). 4. PAO C. V., Coexistence and stability of a competition-diffusion system in population dynamics, J. math. Analysis
Applic. 83, 54-76 (1981). 5. MOTTONI P. DE & SCHIAFFINO A., Competition systems with periodic coefficients: a geometric Biol. 11, 319-335 (1981). 6. SMOLLER J., Shock Waves and Reaction-Diffusion Equations. Springer, New York (1983).
approach,
.I. Math.
Methods
& Applicolions,
Vol.
19. No. 4. pp. 335-344,
1992 0
EXISTENCE PERIODIC
OF GLOBAL COEXISTENCE COMPETITION DIFFUSION
0362-546X/92 $5.00 + .@I 1992 Pergamon Press Ltd
STATE FOR SYSTEMS-f
ANTONIO TINEO Universidad de Los Andes, Facultad de Ciencias, Departamento de Matematicas, Merida 5101, Venezuela (Received 1 October 1990; received in Key words andphrases:
revisedform25 Juty 1991; receivedfor publication 8 November 1991)
Periodic, reaction-diffusion,
global stability, asymptotic behaviour, extinction.
0. INTRODUCTION IN THIS
paper we consider the following system of parabolic equations: U, = P(U) + u[a(t, x) - b@, x)u - c(t, x)u] u, = Q(u) + v[d(t, x) - e(t, x)u - f(t, x)v]
(0.1)
where a , . . . , f: R x $2T, R are positive continuous functions, which are periodic in the time t with period T > 0; Sz is a bounded domain f R”whose boundary 6X2belongs to C2 and P, Q are uniformly elliptic operators in R x fi, whose coefficients are T-periodic in the first variable. When P, Q have the form k(t, x) A(u); where A denotes the Laplacian with respect to the space variables; the system above models the competition between two species, with migration from regions of high population density to regions of low population density. For details see Pao [4] and the references therein. When a , . . . , f and %2 are sufficiently smooth, we shall study the existence and uniqueness of a positive solution (u, u) (U > 0, u > 0) of (0.1) such that v(t + T,x) = u(t,x)
u(t + T, x) = u(t, x), adav
= adav
= 0
in IRx an
(0.2) (0.3)
where a/av denotes differentiation in the direction of the outer normal to X2. We also study the asymptotic behaviour of positive solutions (u, u) of (0. I), which satisfy the following initial boundary value problem: adav
= adav
MO, x) = d(x),
= 0
in (0, 00) x asz
u(O, x) = w(x)
in !2
(0.4) (0.5)
for sufficiently smooth nonnegative functions 4, I,U:d --f R such that a+/av = av/av = 0, in an. To be precise, let us define from now on: B = b/a, C = c/a, E = e/d, F = f/d, and for all bounded functions g: X + R (some nonempty set X); g, = inf(g(x): x E X) and g, = sup(g). In the “stable” case: and FL > CM, (0.6) BL 7 EM j-This work was sponsored by the C.D.C.H.T.
of the Universidad de Los Andes, Venezuela. 335
336
A. TWEO
we prove the existence of a positive solution to (O.l)-(0.3) and we obtain a priori bounds for the positive solutions to this problem, whose components belong to C**‘(IR x a) n C’*‘([R x ai). Actually, our results generalize theorems 1.1 and 4.1 of Ahmad and Lazer [ 11. Further, if (0.6) and C,u&Xu
- C,)(B,
- EL) <
&&FL
-
Cd@,
- &I
(0.7)
hold, then the problem (O.l)-(0.3) has exactly one positive solution. Moreover, this solution is globally asymptotically stable. It is a partial answer to a question in [l]. Remark.
If B, C, E, Fare constant, then (0.6) implies (0.7).
Remark.
If 2 maxlB,C,,
FMEM] I BLFL, then (0.6)-(0.7) holds.
Proof. Notice first that BL L 2E, and so, BL - EM 2 (l/2)8,. Similarly, FL 2 2C, and FL - CM L (1/2)F,. Thus, (0.6) holds and B,F,(F, - C,)(B, - EM) 2 (l/4)(BLFL)* 2 B,C,E,F, > C,E,(F, - C,)(B, - EL). So the proof is complete.
When EL > BM (resp. C, > FM) we study “the domain of attraction of the trivial solution (Uo, 0) (resp. (0, V,))“, where U, (resp. V,) is the unique T-periodic solution to the equation U, = P(u) + u[a - bu] (resp. u, = Q(v) + u[d - fv]), which satisfies the Neumann boundary condition in IRx &X We used these results to improve a theorem in [I], about the extinction of one species. To complete our paper, we exhibit an iterative scheme which allows us to find a positive solution to (O.l)-(0.3), when a > cVo and d > eU,. We shall see that these conditions are implied by (0.6). 1. A PRIORI ESTIMATES In the following, P, Q, R denote elliptic operators of the form
(1.1) where aij, bi: IRx fi -+ R are continuous functions which are T-periodic in the time t and aij = aji. We also assume that there exists E > 0 such that n n C aij(f9 x>titj 2 E C ti’ i,j=l
for all
i=l
in this paper are assumed to be continuous
PROPOSITION1.1. Let w E C* := C*,’ (R x a) fl C”‘(IR x ai) be a nonnegative solution to in m x an. w, = R(w) + gw, aw/av = 0
and
T-periodic
If w # 0 (resp. wL > 0), then there exists (tr, x,) in IRx fi such that w(t,, x,) = w, (resp. w(t,, xl) = wL) and g(tr ,x1) L 0 (resp. g(t,, x1) I 0). Proof.
It follows from the same arguments in [l, lemma 1.21.
Global coexistence state
337
PROPOSITION1.2. Let (u, u) be a nonnegative solution to (O.l)-(0.3) such that u, u E C*. (a) If uL > 0 and uM > 0, then FL - CM I (B,F,
- CMEL)uL
(1.2)
BM - EL 2 (BMFL - C,E,)v,.
(1.3)
(b) If uM > 0 and uL > 0, then FM - CL I (B,F, Proof.
and
- C,E,)u,
We only prove (a). From proposition
BL - EM I (B,F,
- CL E&v,.
1.1, there exist (t, ,x1), (tz, x2) in R x fi such
that 00, , x1) 5 w, 9x1 1% + co, 9x,Mt, w,
3$1 1 4t, 3XMf2
From here, 1 i BMuL + C,u, If B,F,
- C,E,
and 1 L E,u,
# 0 and B,F,
- C,E,
9x1)
, x2) + _m* 3x2h.f. + FL uM; and the proof follows easily.
# 0; we define
E, = (BMFL - C,E,)-‘(FL
- CM)
6, = (&FM - GE,)-‘(FM
- CL)
E, = (B,F,
- EM)
(1.4) - GE,)-‘(B,
6, = (BMFL - &EL)-‘(B,
- EL).
We have the following immediate corollaries to proposition
1.2.
COROLLARY 1.3. If (0.6) holds and (u, u) is a positive solution to (O.l)-(0.3), whose components belong to C*, then E, I u I 6, and E, s u I 6,. Remark.
Assume bL/aM > e,/d, (FM - GM&F,
and fL/dM > c,/a,.
Then, BL > EM and hence
- &CL,) 5 tfM/dL- c,/%M&f,/&
- EIMcJcIIM)
= (aMf,tt - c~&Y(& aMfM- Ed,&) 5 (aMfM- cLdMbLfM - CL%,& since FM d fM/dL and C, 2 CL/a,. [l, theorem 1.11.
From this, it is easy to prove that corollary 1.3 generalizes
COROLLARY 1.4. If B, C, E, Fare constant, then the constant (u, u) = (E,, 6,) (= (6,) E,)) is the unique positive solution to (O.l)-(0.3) in C*. COROLLARY 1.5. Assume FL 1 CM,
BMIEL,
(FL 94~) # (CM, EL)
(1.5)
and let (u, u) be a solution to (O.l)-(0.3) such that u > 0, u > 0 and u, u E C*. Then u = 0. We have a parallel result if: FM I CL, B, 1 EM and (FM, BL) f (CL, EIM).
A. TINEO
338
Proof. Suppose that vM > 0. From (1.2) we get, B,F, - C,E, > 0 and by (1.3), we obtain the reverse inequality. Thus, B,F, - C,E, = 0 and by (1.2)-(1.5), (FL, B,) = (CM, EL). This contradiction proves that v = 0. The other case is proved by the same arguments, and so the proof is complete. 2. UNIQUENESS
In this section, we prove that the conditions (0.6) and (0.7) imply the existence positive solution (u, v) to (O.l)-(0.3), such that U, v E C*. To this end, we need the following result. LEMMA
2.1. Let ((Y,p) be a T-periodic
solution
to the linear
CY~ = P(a) - ga + h/3,
P, =
adav = ap/av = 0 where g, h, k, I are positive
(T-periodic
continuous
problem:
Q(P) + ka - IP in IR x aa
functions)
. sup(k/l)
sup(h/g)
of at most one
defined
(2.1) (2.2)
in iR x 0 and
< 1.
(2.3)
If CY,p E C*, then (Y = p = 0. Proof.
Let us fix positive c 2 maxlaM,
constants
inf(k//)P,)
and
E < g, [l - sup(h/g)
sup(k/l)].
It is not hard to verify that (p(t), o(t)) := cexp(-et)(l, sup(k/l)) is a supersolution to (2.1) and (2.2) such that 01~ 5 p(O) and PM 5 o(O). Hence, cy 5 p and p I o(, since (2.1) is quasi-monotone. Assume now that CX(T,x) > 0 for some (r, x) E IR x fi. Then, for all integers n 2 1; c L 01(r + nT, x) exp(r + nT)
= cly(r, x) exp(r + nT)
+ co
asn+m
and this contradiction proves that (Y I 0. Analogously, /? 5 0. Repeating the proof with (--a, -p), we get, (Y 2 0, p L 0; and so the proof THEOREM
positive Proof.
2.2. If both
(0.6) and (0.7) hold, (24, v) E C* x C*.
solution Suppose
then
the problem
that (u, v), (U, V) E C* x C* are positive (Y = (u/U)
From an easy calculation
- 1,
(O.l)-(0.3)
solutions
is complete.
has at most
to (O.l)-(0.3)
one
and define
p = 1 - (v/V).
we get,
Olt = Po(cr) - bus
+ (a4 V/U)P,
P, =
Q,,(P) + (euU/Va
-fvP
where PO, Q. are elliptic operators which have the same principal part as P, Q respectively. Let E”, E,, 6,, 6, be defined by (1.4). Then, corollary 1.3 and (0.7) imply: sup(cV/bU) and the proof
sup(eU/fV) follows
= sup(CV/BU)
from lemma
sup(EU/FV)
2.1, since (Y,j3 satisfy
I C,B;‘~,E;~E,,.,F~~B,,E;~ (2.2).
< 1;
Globalcoexistence state
339
3.SOMEBASlCCOMPARISONRESULTS In the next section we assume that there exists CYE (0, 1) such that the coefficients of our equations belong to C 01,01”([0, T] x 0) and the coefficients aij(t, x), in (1. l), are differentiable with respect to the space variables and aa,/ax, E C uPn’2([0, T] x a), for all 1 I i,j, k I n. Moreover, we assume JKJ E Cm+*. From theorems 2.3 of [3] and 2.1 of [l] we get the following theorem.
3.1. Suppose that g, h > 0 on R x b. (a) If $I E P’*(fi) is nonnegative and a$/& = 0 in K!, then the problem:
THEOREM
W, = R(W) + w]g - hw],
at-dav
w(0, x) = d(x)
in (0, 00) X Q
= 0
(3.1)
in fi
(3.2)
has exactly a nonnegative solution w = w+,in CF := C 2+a,1+a’2((0, r] x d) n C([O, r] x Cl), for all r > 0. (b) The problem (3.1) has exactly one positive and T-periodic solution w, such that w * E C2+a*‘+u’2([0, r] x Cl) for all r > 0. Notice that, by proposition 1.1, we have inf(g/h)
I w, 5 sup(g/h).
(c) w,(t, x) - w,(t, x) -+ 0 as t --, co, uniformly
(3.3)
on fi; where wm and w, are as above.
In the following, I denotes the subset of C,‘*(a) consisting such that &$/av = 0 in aG. We identify the constant functions From [l, theorem 3.11, we get the following theorem. 3.2. Let ii, _v, 9, 6: [0, co) x fi + R be functions
THEOREM
of all nonnegative in I with [0,03).
functions
r#~
in
C*,‘((O, r] x Cl>n C’*O((O, r] x b) n C([O, r] x b),
for all r > 0,
such that ii 2 u L 0, b L u 2 0, iii, 2 P(U) + ii[a - bii - cg], aii/av
2 0 2 agav
_v, = Q(g) + _v[d - eti - fg] in (0, r] x an
and (u, 3) satisfies the reverse inequalities above. If 4, w E I; ~(0, x) I 4(x) 5 ii(0, x) and _v(O,x) I v(x) 4 U(O, x) in d; then the problem (0. l), (0.4), (0.5) = (0.1, 4, 5) has exactly one solution (u, u) E Cp x Cp for all r > 0. Moreover, g I u I ii and _v I v 5 V in [0, 00) x Q. 3.3. For each @I,I+YE I, the problem (0.1,4, 5) has exactly one solution (u, v) = urn,&), such that U, u E C,YY,for all r > 0. Moreover, if U, (resp. V+) is the unique solution (%,ti, to the equation U, = P(U) + U[a - bU] (resp. K = Q(V) + V[d - fV]) such that U(0, x) = cp(x) (resp. V(0, x) = w(x)) and aU/av = 0 (resp. aV/av = 0) in [0, M) x Sz, then u I U, and u I V, on (0, 00) x i2. COROLLARY
Proof.
It follows
from theorem
3.2, with (ii, _v) = (U4, 0) and (u, V) = (0, I$,).
340
A. TINEO
3.4. If (ul, v,),(u,, u2) are nonnegative solutions to (O.l), (0.4) such that ui, ui E C,? for all r > 0 (i = 1,2) and ur (0, x) 5 1.4~ (0, x) and ~(0, x) I vr (0, x) then, ur 5 u2 and v2 I v, in [0, co) x a.
COROLLARY
Proof. Let US define hi = ~~(0, X) and Ui = U,,, for i = 1,2; where U+ is as above. By [6, theorem 10.11, we get CJ, I U, in [0, 00) x Q, since 4r I &. By corollary 3.3, we have U, 5 U, in [0, co) x 52, and then (ii, _v):= (U,, 0), (g, 13):= (uI , v,), (4, I,V):= (z.+(0, *), v2(0, e)) satisfy the hypotheses in theorem 3.2. The proof now follows from this theorem.
COROLLARY
3.5. Let
(u, v) be a nonnegative solution to (O.l), (0.4) such that U, v E Cp, for all
r > 0, and ~(0, x) 5 u( T, x) and ~(0, x) 2 v( T, x) in Q. Then there exists a nonnegative solution (u* > v*) to (O.l)-(0.3) such that u*, v* belong to C 2+n*1+a’2([0, r] x Cl) for all r > 0; and
as t -+ to
(u(t, X) - u*(t, X)7v(t, X) - v*(t, X)) -+ (090) uniformly on b. Moreover,
(3.4)
U* 2 u and v, I v in [0, 00) x Q.
Proof. For each integer k 2 1, let (Uk, v,J be the solution ‘to (O.l), (0.4) defined by uk (t, x) = u(t + kT, x) and v,(t, x) = v(t + kT, x). By corollary 3.4; u I U, and v 2 vr in [0, 00) x Q, and then ur I u2 I ... and vr 2 v2 2 .... The proof now follows from the arguments in [ 1, theorem 3.21. Remark.
Let (u, v) be as above and suppose that there exists E > 0 such that u(t, x) r E in [0, 00) x Q. If v(t, x) + 0 as t -+ 03 then, v, = 0 and U, is the unique T-periodic positive solution to the equation u, = P(u) + u[a - bu] such that au,/& = 0 in R x dQ. 4,ANITERATlVESCHEME
Let us assume a > cV,,
d>eU,
(4.1)
where U,( V,) is the unique T-periodic positive solution to the equation U, = P(u) + u[a - bu] (v, = Q(V) + v[d - fv]) which satisfies aU,/av = 0 (a&/h = 0) in R x aQ. We shall prove that the following iterative scheme (4.2) vkf = uk(t
+
T, x)
=
Q(vk)
+
uk(t,
x),
vk W vk(t
auk /av = at&/& = 0 provides a positive solution to (O.l)-(0.3). To do this we need the following result.
euk +
- fvkl
T,x)
= vk(f,X)
in R x ai-2
(4.3) (4.4)
Global coexistence state
341
PROPOSITION4.1. Let g,, g, , h: IR x fi -+ IR be T-periodic continuous functions such that h > 0 and g, 5 g, . If Ui E C* is a T-periodic positive solution to the equation: u, = R(u) + u[gi - hu], such that aUi/av = 0 in iR x 80 then, U, 5 U,. Proof.
Let us write V = U/U,,
- 1. Then,
v, = R,(V)
- hU,V + (sr - g,)UJU,
(4.5)
for some second order elliptic operator which has the same principal part as R, and a V/av = 0 in R x an. Assume now that V, < 0 and define IV@,X) = V(t, x) - Vr.. Then W, = 0 and R,(W) - w - hU, W = hUV, + (go - g,)U,/U,, < 0. From [6, theorems 9.6 and 9.121, we get W = 0, and so I/ = V,. By (4.5), 0 I g, - g, = hU,VL < 0 and this contradiction shows that V 2 0. Thus the proof is complete. THEOREM4.2. If (4.1) holds, then problem (O.l)-(0.3) has positive solutions (ii, _v),(u, ii), both of whose components belong to C 2+a,1+a’2(lRx ai), such that: u I u 5 ii and _vI v I z7 for any positive solution (2.4,u) E C* X C* to (O.lHO.3). Proof. We shall prove the existence of two sequences (u,), (uk) of positive functions in C2+a*1+a’2(lRx G) which satisfy (4.2)-(4.4) and u1 2 u2 L ++e; V, I v2 I -.. I V,. Define u, = U,. Then, d - eu, > 0 and by theorem 3.1, the equation
u, = Q(u) + v[(d - eui) - fv] has exactly a positive T-periodic solution vr which satisfies the Neumann boundary condition. By proposition 4.1, u, I V0 and then a - cv, 2 a - cV, > 0. Consequently, the equation: u, = P(u) + ~[(a - cul) - bu] has exactly one positive T-periodic solution u2 which satisfies the Neumann boundary condition. Notice that, by proposition 4.1, u, 2 u2. In this way, we obtain sequences (u,), (u,J with the required properties. From somewhat well-known arguments (see [l, 31) we can prove that (z+., ok) converges to a solution (ii, _v)of (O.l)-(0.3). Notice that _vL u1 > 0. Let u, be the unique positive T-periodic solution to the equation u, = P(u) + u[(a - CV,) - bu] such that au,/av = 0 in R x aa. By proposition 4.1, we get u, r u,, since u, I V,. Thus, ii 2 u, > 0. Assume now that (u, u) is a positive solution to (O.l)-(0.3) such that u, u E C*. Since u 2 u0 = 0, then u, 1 u, and by proposition 4.1, ui I u. By induction, we have u, 1. u and u, I u for all integers n L 1; and so, ii 2 u and _vI u. To end the proof, it suffices to consider the following iterative scheme: ug = 0,
u,t = Q&J + u, id - eu,-, - _Ikl
Unt = P(u,)
+ u, [a - cu, - bu,].
So the proof is complete. Remark. Assume (0.6). From (3.3), we have V,, _ -= l/F’ and so, a - cV, = a(1 - CV,) 2 a,(1 - CM/F’) > 0. Analogously, d - eU, > 0, and thus (0.6) implies (4.1).
A. TJNEO
342
5. ON THE
DOMAIN
OF ATTRACTION
OF THE
Given a nonnegative solution (u,, u,) W(u,, u,) as the subset of I x I consisting
TRIVIAL
SOLUTIONS
to (O.l)-(0.3) we define (the of all points (4, w) such that
(u(r, x) - u*(t, x), u(t, x) - u*(t, x)) + (0,O)
(Cz’,,,O) AND
“stable
(0, I”)
manifold”)
as t + cc)
uniformly in ai; where (u, u) is the unique solution to (O.l), (0.4) and (0.5). It is clear that: (4,O) E W(U,, 0) and (0, I,U)E W(0, V,) for all 4, v/ E I; 4 f 0, I+Vf 0. 5.1. If (&, t+vO)E W(Og , 0) then, result holds for (0, 6).
PROPOSITION
A parallel Proof. w = vO). uniformly On the
(4, I,U)E W(U,, 0) for all 4 L do and I,UI wO.
Let (u, u) ((u,, uO)) be the solution to (O.l), (0.4) and (0.5) (with 4 = &, and From corollary 3.4 we know that u0 I u and u0 2 u and hence, u(t, x) + 0 as t + 03 in d. other hand, if U, is defined as in corollary 3.3, then, u 5 U, and U,#,(&X) - 060,x)
as t -+ to, uniformly
PROPOSITION
in 0. The proof
(4, V) belong Proof. B,4
EL L B,
5.2. Assume
if FL 2 CM and (A,,p(,)
= (~~,a,)
follows
+ 0
now from the following
inequalities:
and FL > CM if EL = BM and define (A,, po) = (0, l/C,) if F’ < CM. If 4, v/ E I and +r > A, and wM < p. then,
to w(Uo, 0).
Let Tbe the set consisting
of all points
(6, I,U)E R2 such that 4 > A,; 0 I I,V< ,u, and
+ C,v
I 1 I EL+ + F,y. If (@, I,V)in T then, (u, V) := ($, I,u), (ii, _v) := (B;‘,
0) satisfy the hypotheses in theorem 3.2 and hence 4 5 u(t, x) 5 l/B, and 0 5 u(t, x) 5 I,Uin [0, 00) x a; where (u, u) is the solution to (0.11, (0.4) and (0.5). In particular, u(T,x) 2 u(O,x) = 4 and u(T,x) I u(O,x) = cy. Let (u,, u,) be given by corollary 3.5. Then, U, 2 4 > 0 and by corollary 1.5, u, = 0 if FL I C,. Assume now that FL < CM. If uM > 0 then, by (1.2), A0 = E, 2 uL 2 4 and this contradiction proves that u, = 0. Consequently, U, = U. and hence, T c W( U, , 0). By proposition 5.1, (A0 , 00) x [0, po) c W(U,, 0) and the proof follows from this proposition.
To obtain
our results about the extinction
of one species, we need the following
proposition.
PROPOSITION 5.3. Let (u, u) be the solution to (O.l), (0.4) and (0.5); 4, v/ E I. If $ + 0 (I,Vf 0) then, u(t, x) > 0 (u(t, x) > 0) for all t > 0 and x E b. In particular, if (u, u) is nonnegative and T-periodic and u f 0 (u f 0) then u > 0 (u > 0).
Proof. We know that U, u are bounded functions on [0, 03) x d (see theorem 3.1 and corollary 3.3). So, there is a constant y > 0 such that y + Q - bu - cv > 0 in [0, 03) x d and thus, P(u) - u, - yu s 0 in [0, co) x b. The proof follows now from [6, theorems 9.6 and 9.121. See [l, proof of theorem 3.21.
Global coexistence state THEOREM 5.4.
343
If FL > CM and BM < EL then, (4, I,@E W(U,, 0) for all 9, w E I; 4 f 0.
Proof. Let 4, w be as above and let k’ = V, be defined as in corollary 3.3. We know that V(t,x) - V,(t,x) + 0 as t -+ 00, uniformly on fi and V, 5 Z$’ < CG’. Consequently, there exists an integer k 2 1 such that V(kT, x) < CG’ for all x E a. Let (u, v) be the solution to (O.l), (0.4) and (0.5) and define
u,(t,x)
= u(t + kT,x),
u1(t, x) = u(t + kT, x).
Then, (ui , II,) is a solution to (O.l), (0.4) and (0.5) such that u1 (0, x) > 0 (proposition 5.3) and vi (0, x) < C,&’ (corollary 3.3) in a. By proposition 5.2, we get (ui (t, x) - U,(t, x), vi (t, x)) + (0,O) as t -+ to uniformly on 0, and so the same holds for (u, u), since U, is T-periodic. The proof is thereby complete. From the change of variables (u, u) + (u, u) we obtain the following theorem. THEOREM
5.5.
If B, > EM and FM 5 CL, then (4, v) E W(0, V,) for all 4, v/ E I; y f 0.
PROPOSITION 5.6. If BL > E,,,,, then W(U,,, 0) = ((4, 0): 4 E I and 4 f 0). (A parallel result holds for (0, I$) if FL > CM.)
Proof. Let T be the set consisting of all pairs (4, r+~)E Rz such that 4 1 0, I,Y> 0 and EM4 + FMw I 1 I BL$ + CL I,Y;and fix (4, I+Y) in T. By the arguments in proposition 5.2, we
have that the solution (u, u) to (O.l), (0.4) and (0.5) is asymptotic, as t -+ 00; uniformly in d; to a nonnegative solution (u,, u,) to (O.l)-(0.3) such that u, > 0. So, T fl W(U,, 0) = $I, and the proof follows from the arguments in proposition 5.2. The results in this section Lotka-Volterra system.
are parallel
6. GLOBAL
to some results in [5], about
the ordinary
COEXISTENCE
From the arguments in Section 5 and theorem 4.1 of [l], we can prove the following theorem. If (0.6) holds and (ii, _v),(_u,V)are as in theorem 4.2, then for each solution (u, u) to (O.l), (0.4) and (0.5); 4, ry E I; 4 f 0, I+Y f 0; and for E > 0, there exists I, > 0 such that THEOREM 6.1.
u(t, x) - & I u(t, x) 5 ii(t, X) + E _v@,X) - 6 5 u(t, x) 5 a(t, x) + 6 if t > t, and x E Q. Remark. By corollary 3.4, we have that (4, w) E W(ii, _v) if 4 I ci(0, .) and y 2 ~(0, a). An analogous remark holds for (u, 5).
From theorems 2.2 and 6.1 we have the following theorem.
344
A. TINEO
THEOREM6.2. Assume (0.6) and (0.7). Then (O.l)-(0.3) has a unique positive solution (u,, uo). Moreover, W(U,, u,,) = ((4, t,~)E Z x I: $J f 0 and w f 0). COROLLARY 6.3. Assume BL > EM, FL > CM, inf(b/e) > sup(c/f ). If a, . . . , f depend only on t, then ii = u, v = B and ii, _vdo not depend on x.
Proof. In theorem 3.1, assume that g, h depend only on t and let w0 be the unique T-periodic and positive solution to: x’ = x[g - hx]. It is clear that w,, is a solution to (3.1) (with R = P) and so W, = w,,; where w* is given by theorem 3.1. So, w, does not depend on x. From this, we have that the same holds for the sequences (u,), (Q) in theorem 4.2, and then (ii, _v)depend only on t. A parallel result holds for (u, 0) and thus (ii, _v),(F, V)are T-periodic positive solutions to the following Lotka-Volterra system: U’ = u[a(t) - b(t)u
-
c(t)v]
u’ = u[d(t) - e(t)24 - f(t)u]. The proof follows from [2, theorem 2.11. Acknowledgement-The
author
thanks
the referee for useful comments.
The shorter
proof
of lemma 2.1 is due to him.
REFERENCES 1. AHMAD S. & LAZER A., Asymptotic behaviour of solutions of periodic competition diffusion systems, Nonlinear Analysis 13, 263-284 (1989). 2. ALVAREZ C. & TINEO A., Asymptotically stable solutions of Lotka-Volterra equations, Rad. Mat. 4, 309-319 (1988). 3. PAO C. V., Positive solutions of nonlinear boundary value problems of parabolic type, J. dvf. Eqns 22, 145-163 (1976). 4. PAO C. V., Coexistence and stability of a competition-diffusion system in population dynamics, J. math. Analysis
Applic. 83, 54-76 (1981). 5. MOTTONI P. DE & SCHIAFFINO A., Competition systems with periodic coefficients: a geometric Biol. 11, 319-335 (1981). 6. SMOLLER J., Shock Waves and Reaction-Diffusion Equations. Springer, New York (1983).
approach,
.I. Math.