on an abstract competition model and applications

Nonlinear Analysis, Theory, Methods & Applicalrons, Vol. 16, No. 11, pp. 917-940. 1991. 0362-546X/91 $3.00+ .OO Printed in Great Britain. 0 1991 Pergamon Press plc

ON AN ABSTRACT COMPETITION MODEL AND APPLICATIONS

PETER HESS

Mathematics Institute, University of Zurich, Ramistrasse 74, 8001 Zurich, Switzerland

and

ALAN C. LAZER

Department of Mathematics and Computer Science, University of Miami, Coral Gables, FL 33143, U.S.A.

(Received 30 April 1990; received for publication 20 September 1990)

Key words and phrases: Periodic competition diffusion system, discrete order-preserving semigroup, global attractor, unstable fixed point, contracting rectangles, fixed point index, principal eigenvalue of periodic-parabolic operator.

INTRODUCTION

LET E, AND E2 be ordered Banach spaces with positive cones Pi and Pz respectively and let the order in both of these spaces be denoted by “5”. Suppose that the coordinates of a point (xi, x2) E PI x P2 are viewed as representing the populations or population density functions of two species in competition with each other and that

F: PI x P2 -+ PI x P2

is a map such that the coordinates of F(x,, x2) represent the population or population densities which have evolved from the state (xi, x2) at a fixed time later. Setting (yi, y2) = F(x,, x2), it is natural to suppose that an increase in x, coupled with a decrease in x2 results in an increase in y, and a decrease in y2 and, symmetrically, a decrease in x1 coupled with an increase in x2 results in a decrease in yr and an increase in yz.

We define an order in E, x E2 as follows: If x,, xi E E,, x2, xi E E,, x1 I xi, and xi I x2, then we write

(x1 7 $1 5 (4 9 x;>.

Although this partial ordering on El x E2 is not the natural one, it is the most appropriate for the abstract competition model we consider since the natural properties assumed for the map F, considered above, imply that F preserves this order.

In the first section of this paper we study the discrete semi-dynamical system defined by iterating a smooth map F: PI x P2 -+ PI x P2 which preserves the order on PI x P2 defined above and satisfies other natural conditions that one would expect to be satisfied for a semi- dynamical system which models competition between two species. For example, it is assumed that F maps each of the sets (PI - (01) x (0) and 10) x (P2 - (0)) into themselves and that the restriction of F to each of these sets has a unique fixed point which is globally attracting (within these sets).

A special case of this problem was considered in [13]. There E, = E2 = R, PI = P2 = [0, co) and F was the Poincart map associated with a two-dimensional system of ordinary differential equations with periodic coefficients which was the classical Volterra-Lotka competition

917

918 P. HESS and A. C. LAZER

system. From the fact that F preserved the order defined above on IR+ x R+, it was observed that both of the numerical sequences formed from the coordinates of the members of the orbit of any point were eventually monotone. From the fact that the sequence of iterates of any point under F was bounded, it was observed that any such sequence converged to a fixed point of F.

In the final section of this paper we apply the theory developed for the abstract model in the next section to a situation in which E, and E2 are certain function spaces and F is the Poincare map corresponding to a system of reaction diffusion equations with Neumann boundary conditions in which the system is periodic in the time variable. This system was previously considered in [ 11. Using the abstract principles developed in the next section for a discrete semi- dynamical system we greatly improve some results of [l]. This is accomplished by replacing certain pointwise inequalities by integral inequalities which have implications for the discrete system.

1. THE ABSTRACT FRAMEWORK

For definitions pertaining to terms used in this section we refer the reader to [4, 10, or 161. In this section for k = 1,2, Ek will denote an ordered Banach space and the cone defined by

the ordering in Ek will be denoted by Pk. If xk and XL are two elements in Pk , then the notations xk s XL, xk < x;, and xk Q x; will mean that xl - x, E Pk, x; - xk E Pk - (01, and XL - xk E Int Pk, respectively, where Int Pk denotes the interior of Pk (which is assumed to be nonempty). The norms in El and E2 will both be denoted by I].[[. If x, and x; are in Ek and xk I XL, then [xk, xi] will denote the set of z E Ek such that xk 5 z 5 XL.

We consider E, x E2 to be an ordered Banach space by defining the algebraic operations in the usual way, by defining the norm of (xi, x2) E E, x E, by 11 (x1, x,) 11 = 11x1 II + 11x2 11, and with an ordering defined as in the Introduction, i.e. P = PI x (-P2).

We assume that F: PI x P2 4 P, x P2

is a C’ mapping and the following assumptions are satisfied: (Al) F is order-compact: If xk E Pk for k = 1,2, then

F(]O, x,1 x 10, xzl)

is a relatively compact subset of Pi x Pz.

(A2) Letf,:E,xE,+E,,k= 1,2bedefinedby

F(x, ,x,1 = (fi(x,,xz),fi(x,,x,)).

Ifx,,x;EPkfork= 1,2,O<x, sx;,O<x;~x,,and(x,,x,)#(x;,x&then

and

(A3)

and

0 -e _&(x1, x2) 4 fi(4 * x;),

0 Q f2W 3 xi) -e f&1 9 3).

fi(O, x2) = 0, x2 E p2,

f2h 7 0) = 0, x1 E PI.

(1.1)

(1.2)

Abstract competition model and applications 919

(A4) Let 13~ : Pk + Pk , k = 1,2 be defined by

e,(x) = “6(x, O), x E 5,

@z(x) = J-2(0, x)9 x E P2.

For k = 1,2 there exists a unique & E Pk - (O] such that 0,(&) = &. Moreover, if 0 c t < 1, then r& Q &(T&) Q rZ,, and if 1 < r, then & Q &Jr_&) e r&.

(A5) If D1fi(O, .&) E L(E, , E,) and D2fz(g1, 0) E L(E, , E,) are the derivatives of the maps PI + PI, defined by x -+ fi(x, &), and P2 -+ Pz defined by x 4 f#i, x), respectively, at x = 0, then

and

&.&@r, WP2 - KN C Int P2.

Remarks. 1. It follows from (Al) that F is compact, i.e. F maps bounded subsets of PI x P2 into relatively compact subsets of P, x Pz. To see this, for k = 1,2 let zk E Int Pk. There exists 6 > 0 such that if z E Pk and llzll < 6, then i& - z E Pk. If B is a bounded subset of PI x Pz, then there exists r > 0 such that if (xi, x2) E B, then llxiI/ < r and Ilx,II < r. It follows from the above that if (xi, x2) E B, then for k = 1,2

Consequently 0 5 x, 5 (r/&k.

F(B) E NO, W4z,l x LO, (r/4z21),

so, by (Al), F(B) is relatively compact. On the other hand, since, in general, order intervals are not bounded, (Al) is stronger than the assumption that F is compact.

2. From (A2) and continuity of F, it follows that F is order-preserving with respect to the order defined on El x E,, i.e. if (x,, x2) and (xi, xi) are in P, x P2 and (xi, x2) I (xi, x;), then

F(x, , xd 5 F(x; , x;h

Moreover, F is strongly order-preserving on (PI - (0)) x (Pz - (01): (x1, x2), (xi, xi) E

(8 - PI) x (Pz - Pl), (x1, x2) 4 (xi, xi), (x1, ~2) # (xi, xi) imply F(x,, x2) 4 F(xi, 4). Since F is compact, the following abstract formulation of the method of subsolutions/super- solutions, and monotone iteration (e.g. [4]) applies: if (x,, x2), (xi, xi) E P, x P2, (x1, x2) I (xi, xi), (x,, x2) 5 F(x,, x2), and F(x;, xi) 5 (xi, xi), then there exist (Y,, y2) and (zi, ZJ in PI x P2 such that

(x1 9 x2) 5 (Yl 9 Yz) 5 (Zl , z2) 5 6%) x9,

F(Y,, Y2) = (Y,, yz), and F(zi, zz) = (zi, zz). If (xi, .Q) 5 (ui, u,) 5 (xi, x9 and F(u,, u2) = (ui, u,), then

(Yl, Yz) 5 (u1, u,) 5 (z, > zz).

Moreover, if Fck) is defined by F (‘) = F, Fck+‘) = F 0 Fck’ then lim Fck’(x,, xz) = (yi, y2) and k-m

lim Fck’(x;, xi> = (zl , z2), and the sequences are monotone increasing, respectively decreasing.

%I; will be used several times in what follows. 3. The intuitive basis for the assumption (A2) when the coordinates of a point (xi, x2) E

P, x Pz are viewed as representing the populations or population densities of two species in


competition with each other has been explained in the introduction. The fact that “4” rather than “<” occurs in (1.1) and (1.2) when it is assumed that 0 < x1, 0 < xi, (xl, x2) s (xi, xi), and (x1, x2) z (xi, xi) is attributed to diffusion of population densities.

4. It follows from (A2) that

F((P, - {O]) x (PI - (0))) c Int P, x Int PI.

5. Assumptions (A3) and (A4) may be interpreted as saying that once one of the species is extinct, it remains extinct, and that in the absence of one of the species, there is a unique nonzero equilibrium state of the other species such that it increases in time if it is nonzero and below this state and it decreases in time if it is above this state.

6. Let (lst, _%?J E (Pt - (0)) x (P2 - (0)) be a fixed point of F. Then

0 & Xi Q 8, and 0 4 z.j % 22

by (A2) and (A4). Indeed

Then x1 Q 0,(~i) and e,(r_&) s ~2, for r >- 1. There exists t, z 1 such that K, I t,%?,. By monotone iteration we get a fixed point of 0i in [Zi, TV,?,]. By uniqueness, this is .?i. Hence

The claim for x2 follows similarly. 7. That D,f,(O, &) P, E P, and D,f&i?i, O)P, E Pz follows from the other assumptions.

Indeed, from remark 2, it follows that if z E PI and s E (0, a), then

The stronger condition (AS) turns out to be satisfied in all applications we consider. From remark 1, it follows that the maps PI -+ PI and Pz -+ P2 defined by z + fi(z, 1;2)

and z. -, f#i, z) are compact. Therefore, by a standard result [lo], the linear operators D,f,(O, Q: E, -+ E, and D2 f#, , 0): E2 4 E2 are compact. It follows from assumption (AS), and a version of the Krein-Rutman theorem (see [4 and lo]) that there exist uk, k = 1,2, such that vk E lnt Pk, IIvk)) = 1, and

Qfi(O, %)u, = A, Ul, (1.3)

Dz f2(& f Oh = 1, v2 9 (1.4)

where A1 > 0 and ;12 > 0 are the spectral radii of D,f,(O, &) and D,f#i, 0), respectively.

1.1. The compressive case

We say that the mapping F: PI x P2 + PI x Pz is compressive if there exist (x* ,y*) and (x*,y*) in PI x Pz such that

0*x* 5x*, (1.5)

o<y*zGy*, (1.6)


m* 9 Y*> = (x* 3 Y*), (1.7)

F(x*,Y*) = (x*,Y*), (1.8)

and such that the order-interval I = [(x* , y*), (x*, y*)] attracts (Pi - (0)) x (P2 - (01) in the order-topology. This means that, given any zi, z[, wi, wj E PI and any z2, z;, w2, w; E Pz satisfying

z1 Q x,, x* * z;, (1.9a)

z2 ey*, Y, + z;, (1.9b)

and 0 < w, I w;, 0 < w, I w;, (1.10)

there exists an integer N such that m 1 N implies

F’“‘(k, w;1 x [wz, %I) c [ZI, z;1 x [z2, z11.

The main result of this section is the following theorem.

THEOREM 1.1. In order that F be compressive, it is necessary that

If

then F is compressive.

A, L 1, A2 2 1.

I, > 1, A,> 1,

(1.11)

(1.12)

The proof of the first assertion will follow from the following proposition.

PROPOSITION 1.2. If

A,< 1, (1.13)

then there exist a number si > 0 and j2 E P2, with j2 Q &, such that if (xi, x2) E PI x P2 and

0 5 xi I sivt, E2 5 x2,

then lim F’“‘(x,, x2) = (0, _f2).

m-m Similarly, if

22 < 1, (1.14)

then there exist s2 > 0 and 9i E PI with jJi Q L?,, such that if (xi, x2) E PI x P2 and 9, I x,, 0 5 x, I s,v,, then

lim F’“‘(x,, x2) = (fl, 0). r?l+cc

Proof. We prove only the first statement, the proof of the second being entirely analogous. Assuming (1.13), we first show that there exists a number 6 > 0 such that if 0 I R, I au,, (1 - d)Z2 I x2 I &, and

F(X, , %2) = @I, %2),

then R, = 0 and x2 = &. Assuming the contrary, there exists a decreasing sequence of numbers {S,]Op with 6, > 0 for m = 1,2, . . . and a corresponding sequence (x,, , x,,) E PI x P2 such


that 6, -+Oasm-+ co,

R&z,, %,) = @m, 3 %J> (1.15)

(1 - 6,)& I Xmz 5 2.2 ) (1.16)

0 I &, 5 &VI, (1.17)

and (X~,,X,_) # (0, -12,). Since C& is the only element z in Pz - (0) such that F(0, z) = (0, z), it follows that x,_ # 0 for m large. Therefore, since fr(0, x~,) = 0, we have that

I

%, = .W,n, , %J = i

D, .M%,, , +%,&%, dt. (1.18) 0

Since (.I&, , x,J is in [0, 6, vr] x [0, a,] for all m 2 1, it follows from (Al) and (1.15) that we may assume, without loss of generality, that the sequence ((x~, , xm,)]~ converges. From (1.16) and (1.17) we then have

lim (&, , em,) = (0, %I. (1.19) m-m

Form = 1,2, . . . . let w, = x~,/]~x~,II. From (1.18), we have

W, = Q A@, %)w, + ! ‘(Q _Ut%, 9 %n,) - 4 fi(O, %2))wn dt. (1.20) ,O

Since D, fr(0, &) is a compact linear operator and (1 w,II = 1 for all m, we may assume without loss of generality that (Or fr(O, ,Qw,]y converges. Hence, from (1.19) and (1.20) we see that lim w, = w E P,, II wI( = 1 and D, fr(O, &)w = w. But, since ,I1 is the spectral radius

m+m of Qf,(O, .&), this contradicts the assumption (1.13). This contradiction shows that there exists 6, 0 < 6 < 1, such that (0, 2,) is the only fixed point of F in [0,6u,] x [p2, a,] where j$ = (1 - 6)&.

Let y be chosen so that 13, < y < 1. Since D, f,(O, &)ur = A, vr 4 yv, , we may assume 6 is so small that Drfr(O, j&r 4 yu,. We infer the existence of (Y > 0 such that if zr E E, and

lIzlll < 01, then Q “fl(O, 92h + Zl -=s w1* (1.21)

By making 6 smaller if necessary, we may assume that if 0 5 s 5 6, then

ll~lfi(~~1,92) - Qfia9*)ll < 01. (1.22)

Since j$ = (1 - 6)g1 a &, it follows from (A4) that jz 4 f2(0, p2). We can therefore choose sr so that 0 < sr I 6 and

92 9f&,u,,92). (1.23)

Sincefr(O,jJ = 0 we have that

fi(sr Ul, 92) = D, ha Jwl Ul) + Sl z17

where

zr = n ‘(Q fi(ts, u1,9d - Q fi(O, _&))u~ dt. I ,O


Since s1 5 6, it follows from (1.22) that [Izl (1 < CX. Therefore, from (1.21) and the above, we have that

fi(% UlY 92) 4 s1 Vl Q s1 Ul. (1.24)

In terms of the ordering defined on E, x E2, we see that (1.23) and (1.24) imply

W, VI 9 92) = (f&l Ul, 9A .m, Ul, 92)) 5 (Sl 4 9 92). (1.25)

Since (0, rZ,) I (.sl u,, j$) and F(0, &) = (0, &), it follows by the iteration theorem applied to the order interval [(0, j$), (s, u,, jQ] in E, x E2 (see remark 2) and (1.25), that

lim FCm’(s, ul, j2) = (X J) 9 9 m-m

where (0, &) i (x, Jo) I (sl ul, j$) and F(x, J) = (X, J). Therefore, since 0 I R 5 s1 v1 5 6v, and j$ = (1 - 6)& 5 J I &, it follows from what was shown above that (x, j) = (0, -12,).

Now let (x1, x2) be any point in PI x P2 such that 0 I x1 I s1 u1 and j$s x,. Let z > 1 be so large that x, < T&, which is possible since _& E Int Pz . According to (A3) and (A4), F(0, zi2) = (0, z), where & Q z 4 r$. Since (0, &) I F(0, T&), (0, i2) = F(0, X$2>, and (0, &) I (0, &), it follows from the iteration theorem applied to the order interval [(0, T&), (0, &)I that there exists (2, J) in this order interval such that lim F@‘(O, &) = (x, ~1) and F(x, J) = (x, 9).

*Am Since, by (A3), this implies that K = 0, & 5 y, and jj = &(O,J), it follows from (A4) that lim F’“‘(0, ~2~) = (0, i$). Now (0, T&) I (xl, x2) I (s, q, j$) implies that

m-)c@

F’“‘(0, ~2~) 5 F’“‘(x,, x2) I F”‘$, ul, j2). (1.26)

By (Al), some subsequence of the sequence (F’“‘(x,, x2)]: must converge and, according to (1.26) and the above, the limit of such a subsequence must be (0, j&2). Hence

lim Fcm)(xl, x2) = (0, _&). m-m

This proves proposition 1.2.

If there exists a relative neighborhood of (gl, 0) ((0, 2,)) in PI x Pz such that for any point (x1, x2) in the relative neighborhood,

lim F@‘(x,, x2) = (a,, 0) lim F@‘(x,, x2) = (0, _f$) , m-m m-m >

then we say that (i1, 0) ((0, ~$2)) is attracting. Obviously F cannot be compressive if either (gl, 0) or (0, j&2) is attracting. If A, < 1 and s1 and j2 are as above, then since 0 -+ u1 and j$ 4 .$ , any point (x,, xz) in PI x P2 sufficiently close to (0, Q will satisfy x1 4 s1 u1 and j$ Q x,, so lim Fcm)(xI, x2) = (0, A$). Therefore, 13, < 1 implies that (0, Q is attracting and similarly,

m+m A2 < 1 implies that (Z1, 0) is attracting.

To prove that the strict inequalities (1.12) are sufficient for F to be compressive, we first prove the following lemma.

LEMMA 1.3. If A1 > 1, (1.27)

then given y, with 1 < y < al, there exist s, > 0 and y2 E P2 with ,+& Q Jo, such that 0 < s < s1


implies that

(YSU, 3 Pd 5 mh 3 92). (1.28) Similarly, if

A,> 1, (1.29)

then given y, with 1 < y < &, there exist s2 > 0 and 9, E PI with ii a pi such that 0 < s 5 s2 implies that

F(Y, I %> 5 01, wd. (1.30)

Proof. We only prove the first assertion. Let (1.27) hold and let y be chosen so that A, > y > 1. Since yu, and 11zi(( < 01, then

Let 6 > 0 be so small

& A, vi = D, fi(O, jZ2)u1, there exists a number (Y > 0 such that if zi E E,

yu, 4 D, fi(0, .%)q + z1.

that if _ijz = (1 + 6)& and 0 5 s 5 6, then

ID,f,(%, Pz) - Qfi(OV %)I1 < cY*

(1.31)

(1.32)

If 0 < s I 6, it follows that, sincef,(O,g,) = 0,

s

1

Ah, ~2) = Q _hW,, ~21~~1 dt 0

where

Since, according to (1.32), ljz,II c CY, it follows from (1.31) that for 0 < s I 6,

vu1 Q _h(w 3 J2). (1.33)

It follows from (A4) that Z2 Q fi(O, jjji2) e p2. Therefore, we may choose a number s1 I 6 such that if 0 5 s I si, then

f&u1 9 Pz) Q -72. (1.34)

From (1.33) and (1.34) we see that oJ.sul, p2) 5 F(sv,, y2) for all s with 0 < s I si. This proves the lemma.

LEMMA 1.4. Assume that A, > 1 and let 1 < y < I,. If si > 0 and jj2 E P2 are as in the state-

ment of the previous lemma, then (x, 9) E (PI - (0)) x (P2 - (0)) and F(x,_~J) = (.Y, y) imply that si vi Q R, J 4 jjz.

Similarly, under the assumption 1 < y < &, if s2 and it are as in the previous lemma and (x, p) is as above, then K Q it, s2 v2 Q J.

Proof. We know from

Now choose s > 0 so integer m 2 1 such that

remark 6 that

J@.&Zjjyz. (1.35)

that sui 4 X and assume that s < s,. Since y > 1, there exists an

Y “-ls I si < yms. Assuming that k is an integer with 0 5 k < m


and that

if follows from the previous lemma, with s replaced by yks I sr, that

(yk+isv,, 92) 5 F(yksu, ) jq I F(k+l)(SvI, jq. (1.37)

Since (1.36) holds for k = 0, it follows by iteration that (yms~r,YZ) 5 F(m$sv,, _Q. From our choice of s and the above, (svr , jQ I (x, y), so by (A2), F(m)(~~,, Y2) 5 F’“)(x, J) = (x, J). Therefore, s1 v1 & Y’%u~ I R and the first part of the lemma is proved. The second part follows similarly.

Proof of theorem 1.1. That conditions (1.11) are necessary for F to be compressive follows from proposition 1.2. Suppose that the strict inequalities (1.12) hold and let s1 , s, , jjl E PI and Yz E Pz be as in the statement of lemma 1.3. Let s > 0 and s’ > 0 be chosen so that

0 < s I sr, 0 < s’ I s,, (1.38)

su1 a Jl, s’vz e 92 . (1.39)

We have that (svt , JJ I (Jo, S’Q) and according to lemma 1.3, (svt, JJ 5 (ysv,, jj2) I

F(sq, _h) and F(_h, S’Q) 5 (PI, s’w) 5 (_h , s’Q). Therefore, by the theorem on monotone iteration applied to the order interval [(svi, pz),

(yr , s’v~)], we infer the existence of (x* , y*) and (x*, Y*> such that

(sv1, Y2) 5 (X* , Y*) 5 (x*, u*) 5 (Y, 9 s’uz),

lim F”‘%t+, Pz) = (x* , Y,), m-m

lim F’“‘(g,, s’uz) = (x* y*) 7 9 m+co

and (x, p) E [(svt , jQ, (Jo, s’Q)] and F(x, J) = (x, J) imply that

(x* , Y*) 5 (-% P) 5 (x*, y*).

We claim that the pairs (x, , y*) and (x*, y*) are independent of the choices of s and s’ satisfying (1.38) and (1.39). Indeed, let s, > 0 and $, > 0 replace s and s’, respectively, in these inequalities and let (x0*, yO*) 5 (xg*, yg) be the fixed points of F obtained by monotone iteration starting with (sO v,, y2) and (pr , .s; vz), respectively. According to lemma 1.4

and

and so

Therefore (so, > P2) 5 (xg*, Yo*) 5 01, s’v2).

(x* ,Y*) 5 ($,Yg*) 5 (x*,y*) (1.40)

and the same argument shows that (1.40) holds with (xg*, y;) replaced by (xO,,y,,*). Consequently

(x*, Y*) 5 (x0* 9 Yo*) 5 (xg*, Yo*) 5 (x*, y*).


Interchanging the roles of s and sO, and s’ and s;, in this argument shows that

(%*,Y0*) 5 (x*,y*) 5 (x*,y*) 5 (Xg*,Yo*).

Hence (x0*, yO*) = (x, , y,), (xi, yg) = (x*, y*) and the claim follows. To show that F is compressive let zi, z;, w,, w; E PI and zz, z;, w2, w; E P2 be chosen so

as to satisfy (1.9a), (1.9b) and (1.10). Choose multiples U, of 2, and u2 of & so that

w; 4 u,, 2, Q ur (1.41)

w; Q u2, $ Q u?_. (1.42)

According to (A3) and (A4) and the conditions j?i Q pi, & -G y2, there exists an integer Ni such that

FcN’)(u,, 0) = (u;, 0) ,

FcN”(O, u2) = (0, u’) 2 7 where

5?i eu; Q p,,

Let (xi, x2> E: [wl, t-v;] x [w2, w;]. Since, according to (1.41) and (1.42), (0, u,) 5 (xi, x2) 5 (ui, O), it follows that for such (xi, x2),

@,_h) 5 F(N1)(~,, x,) s 01, 0).

Therefore, since (xi, x2) E [w,, w;] x [w,, w;] implies that

FcN1)(wl, w;) 5 FcN’)(x,, x2) 5 FcN1)(w;, w,),

and since (A2) and (1.10) imply that FcN1)(wl, w;) and FcN1)(w;, w,) are in Int PI x Int P2, we infer the existence of numbers s and s’ such that 0 < s 5 si , 0 < s’ 5 s2, and

b,,J2) 5 FcN’h x2) 5 @til, ~‘~21

for all (xi, x2) E [wi, w ;] x [w2, w;]. From this, the facts that lim F’“‘(sv,, p2) = (x*, y*) m*m

and lim F(m)(J,l, S’UJ = (x*,y*), and from conditions (1.9a) and (1.9b), we see that there m+m

exists an integer N 2 Ni such that

F(‘%w,, w;l x 1~2, w;l) c [z,, z;l x k2, z;l

for all m 2 N. This shows that F is compressive and the theorem is proved.

Remarks. 8. If A1 > 1 and a2 > 1, it follows that Z is globally attracting (in the original norm), i.e. that for each (xi, x2) E (PI - {O)) x (P2 - {O)), Fcm)(x,, x2) -+ Z as m -+ co. Indeed, if (xi, x2) E (PI - {O)) x (Pz - (OJ), there exists Ni such that

(svr , p2) 5 F@‘l) (XI, x2) 5 01, s’u2).

Hence the sequence (FcNlem) (x1, x2)): lies in the relatively compact set F([su,, yl] x [s ‘u2, p2]). Let U, be the open e-neighborhood of I. We claim that there exists mE E Z N such that


F(N1+m)(xl, x2) E U, for all m 1 m,. If not, there is a subsequence mk -+ 00 with

F(Nl+mp’(X1) XJ $ u, and F(Nl+m*)(Xr, x2) -+ (z?, 2;).

Of course (zf, 2;) $ U,. But

(x* ) y,) + F@qsv,, jq 5 FcNl+yXI, x2) I P@(J~, S’Q) + (x*, y*> (k + 00)

implies that (z:, zz) E I. This contradiction proves the claim. 9. It follows from [7, theorem 31 that there is at least one stable fixed point in I.

1.2. Extinction

We say that (xi, ~~2) is a coexistence state if F(x,, x2) = (xl, &) and X, > 0, X~ > 0 ((A2) implies 0 Q &, k = 1,2).

We have the following theorem.

THEOREM 1.5. If Ai > 1 and there exists no coexistence state, then (_?r, 0) is globally attracting. If Az > 1 and there exists no coexistence state, then (0, a,) is globally attracting.

Proof. Assuming that I, > 1, let A, > y > 1 and sr and yz be as in the statement of lemma 1.3. Let s > 0 be chosen so small that s < s1 and sur < g2,. Choose yi = rgi with 7 > 1. According to lemma 1.3, @vi, p2) 5 (syu,, J2) I F(su,, j%J and according to (A3) and (A4), F(y,, 0) I (yt , 0). Since (.svt , y2) 5 (yr , 0) it follows from the iteration theorem applied to the order interval [(sur , jQ, (yr , 0)] in El x E2 that lim F@‘%v,, _F$) = (2, Jo) where F(X, _V) =

m-co @,p) and (sur,yJ I @,p) 5 (yr, 0). Assuming that there are no coexistence states, it follows that, since sur 5 K, p = 0. Hence, by (A4), (X, 0) = (ii, 0). If (.svr , jQ I (x, y) I (yr , 0), then for m 2 0

F’“‘(.sv,, jQ I F’“‘(x, y) I F@“(y,, 0)

and since FCm’(y,, 0) -+ (2, 0) as m + 00 (see proof of proposition 1.2), it follows that lim FCm’(x y) = (.I? 9 0) , . m-+m

In the general case, if (x,y) E (PI - (0)) x (P2 - {O)), then, if yr is as above, the argument used in the proof of theorem 1.1 shows that there exists an integer Nr such that

(O,P,) 5 F’N”(x,~) 5 (~1, 0).

Therefore, since FCN ) 1 (x, y) is an interior point of PI x P2, there exists s with 0 < s < s,, such that (sr vi, y2) I FCN1)(x, y) I (yl, 0). Then, from the above, it follows that F@'(x, y) + (c?~, 0) as m + 0 and the proof is complete.

1.3. An unstable coexistence state

Finally, with an extra assumption, we prove that the conditions J.r < 1 and A2 < 1 imply the existence of an unstable coexistence state.

We note that, since fk(O, 0) = 0, k = 1,2, and fk(xl, x2) 2 0 for (xi, x2) E P, x P2, it follows that Dk fk(O, O)P, E Pk. Also since, according to (A4), 0 is an unstable fixed point of ak for k = 1,2, we must have that the spectral radii of the linear operators Dk fk(O, 0) = Dkflk(0)


are at least equal to 1. Our extra assumption, which is satisfied in applications, is slightly stronger than these implications.

(A6) For k = 1,2, Dkfk(O, O)(P, - (0)) C Int Pk and the spectral radius of Dkfk(O, 0) is larger than 1.

THEOREM 1.6. If, in addition to (Al)-(A5), assumption (A6) holds, and

A,< 1, k= 1,2, (1.43)

then there exists an unstable coexistence state.

In the proof we use the fixed point index i(F, X, U) of the compact map F relative to the closed convex subset X = PI x Pz of E, x E2, with U being the relatively open subset U = B, (0) n X, BR (0) = [(x1, x2) E E, x E2 : 1) (x1, x2) 11 < R ). The radius R will be fixed as follows.

LEMMA 1.7. For R > 0 sufficiently large, i(F, X, U) = 1.

Proof. Consider the homotopy H(t; x1, x2) = tF(x,, x2) + (1 - t)(&, 11J, 0 I t I 1. We show that for R sufficiently large, (xi, xz) f H(t; x1, x2) for all (xi, x2) E aU, 0 I t 5 1. Since (gr, 2,) E U, this then proves the claim.

Suppose H(t; x1, x2) = (x1, x2) for some (xi, x2) E X, 0 I t I 1. Then

x1 - tfi(xl, x2) = (1 - 0% x, - &(x, , x2) = (1 - t>.fz . (1.44)

If t = 0, (xl, x2) = (gl, $2). Next assume 0 < t 5 1 and xk E Pk - (0) (k = 1, 2). Then

0 < xi = gi(x,, x2) + (1 - t)xl

Q tfl(x,,o) + (1 - t&q.

By (A4), r,%?i %-ffi(r5?i, 0) = tfl(~21, 0) + (1 - t)f,(rZ,, 0) 1 tfl(dl, 0) + (1 - t)2, for all t > 1. We claim that xi 5 Zi. Indeed, consider the mapping &(zJ = tfl(zl, 0) + (1 - t)i?, : PI --+ PI and let t* = inf (7 > 1 : 7Zl 1 xl). Assuming that 7* > 1, we obtain

xi Q 0,(x,) _= Q,(r*a,) * 7*q,

contradicting the definition of r*. Thus 7* = 1 and xi i gl. Similarly x2 I & follows. If, e.g. x, = 0, (1.44) implies that t = 1 and x1 = 0 or gi. In any case, (xi, x2) is contained in the relatively compact set

which gives a bound on 11 (x1, x2) 11. Choosing R larger, the lemma follows.

LEMMA 1.8. The fixed point (0,O) has local index i(F, X, (0,O)) = 0.

Proof. We show that (xi, x2) # H(t; x1, x2) for all (xi, x2) E X with 11(x1, xz)ll = E > 0 small enough, for all 0 5 t 5 1. Since i(H(0; ), X, (0,O)) = 0, this proves the lemma.


Suppose, to the contrary, that there is no such E > 0. Then for m = 1,2, . .., we find t, E LO, 11 and (xml, .M E X with Il(xmI, x,&II -+ 0, such that (xml, x,,,J = W,; x,,, x,,d. Setting

we infer that

(Wnl~

1 W?Iz) = ,,(x*l, x*2)(( kn17 Gz2)~

(%izl~ w,J - t,,JWA Ww,n,, w,nJ = 1 - t,

IIk?ll9 %?dII (-6, %2) + o(l).

Since the left-hand side is bounded in E, x E, , we conclude that t,,, -+ 1, and that we may pass to a subsequence such that

In the limit, (w-i, ~02) - DW’, Wwco,, wm2) = c&f,, A?~). In particular, wml - Dlfi(O, O)w,, = cdl with w,i _ > 0. Since spectral radius Dlfi(O, 0) > 1, the theory of strongly positive compact linear operators implies that (Y = 0 and w,i = 0. Thus wm2 - D2f2(0, O)w,, = 0, wa2 2 0. By the same argument, wm2 = 0, which contradicts II(wmi, w,,)I( = 1.

LEMMA 1.9. The fixed point (2, , 0) has local index i(F, X, (A!~, 0)) = 1.

Proof. Here we consider the homotopy I?(t; x1, x2) = tF(x,, x2) + (1 - t)(gl, 0) and claim that (xi, x2) # l!?(t; x1, x2) for all (xi, x2) E 8?,(~, , 0) (boundary relative to X, E > 0 sufficiently small), for all 0 5 t 5 1.

Assume that there are sequences (x,, , x,& -+ (gI, 0), x,,,~ 1 0, (x,~, xm2) # (A?~, 0), and t, E [0, 11, such that (x,,, xm2) = I?(&; x,,, xm2). We observe that xm2 # 0 for large m. Indeed, x,,,~ = 0 implies that x,, = t,f,(xml, 0) + (1 - t,&fl andx,, # 2i. Hence t,,, # 0. For each m we consider the mapping &,(z,) = t, fi(zl, 0) + (1 - t,,J2l : PI + PI and claim that its only positive fixed point is 3?i, thus reaching a contradiction. In fact, x,i 5 g1 as in the proof of lemma 1.7. Since xmi %- 0, we have 0 < T, = sup(t < 1 : d1 I x,,J. If t* < 1, x,, = &Jx,,) 2 &&,Z,) = t, f,(q,&, 0) + (1 - t&Z, s tmt& + (1 - t&& = T,_$ by (A4), contradicting the definition of T*. We get t, = 1 and L& 5 x,i, thus x,i = ~?i. If w, = x,,/IIxm211 for large m, then

W” = tmDzfk%v ON,

i

1

+ L P2 f2(x,l, sz2) - D2 fd-% 3 ON wm d.s. 0

By the same type of compactness argument used before, there exist w, E P2, (I w,(( = 1, and t, E [0, 11, such that w, = tmDZ f2(&, O)w,. Since this contradicts the assumption that the spectral radius of D, f2(g,, 0) is less than 1, the claim is established.

In the same way we prove that i(F, X, (0, Z2)) = 1. By lemmas 1.7-1.9 and the additivity of the fixed point index, the existence of a further fixed point in U follows. This must be a coexistence state.

In order to show that there is in fact an unstable coexistence state we follow an argument developed in [7].


As in the first part of the proof of proposition 1.2 we see that (2,) 0) is the isolated fixed point of F in X. Since all fixed points lie below (gl, 0) by remark 6, and since the set of fixed points is compact, by Zorn’s lemma there is a maximal nontrivial fixed point-say (x1, Q-below (gl, 0). Let Z be the order interval [(zl, x,), (gl, O)]. It is clear that i(F, Z, Z) = 1. Since i(F, Z, (gl, 0)) = 1 and there is no third fixed point of F in Z, i(F, Z, (xl, x22)) = 0. Considering the homotopy R((t; x1, x2) = tF(x,, x2) + (1 - t)(..~~, _T~,), in any &-neighborhood of (x, , ~~2) in Z there exist thus (x,,, x,*) E Z and t, E [O, 11 such that (x,~, xEz) = R(t,; xc*, x,J. Since

(X1, X2) 5 (G, xcz) and (X1, %) f (x,~, x,~), (X1, Q = F(X,, X2) 4 F(x,, , x& (as F is strongly order-preserving in Int PI x Int P2). We get

(x,1 > x,d = t,F(x,l, x,z) + (1 - t,)(xl, -%2) 4 W,, > x,z)

(t, = 1 is impossible since there is no further fixed point in Z). Iteration implies that

Fm(xE, , x,J -+ (.fl, 0) as m + 00. Thus (x1, z2) is unstable.

2. APPLICATION TO THE STUDY OF PERIODIC COMPETITION DIFFUSION SYSTEMS

For explanation of terms and notations used in this section see [ 1, 31. We consider the system of parabolic P.D.E.s on fi x R

U, = k1 Au + u[a - bu - cu] (2.1)

u, = k2 Au + v[d - eu - fv]

subject to the Neumann boundary conditions

at4 0,

av

an= an’ 0 on asz x R. (2.2)

Here Q is a bounded domain in R”, whose boundary an is a C2+a manifold for some CY E (0, l), and the functions a(x, t), . . . , f(x, t) are assumed to be T-periodic in t on d x R where T > 0. Further b, c, e, f are assumed to be strictly positive on d x R. We assume that k,(t) and k2(t) are positive T-periodic functions of t of class Ca’2(R), functions a, . . . , f are of class c Q,~‘2(fi x R).

To apply the abstract theory to the study of solutions of (2.1), (2.2) which satisfy the additional condition

u(x, t + T) = u(x, t), v(x, t + T) = v(x, t) (2.3)

for (x, t) E (a x I?), we let IE be the Banach space Ci which is the subspace of functions 4 of C’(a) satisfying the boundary condition a$/& = 0 on an. The set P of nonnegative functions of [E is a solid cone and we consider IE to be ordered by this cone. We order the product space [E x IE as in the previous section.

Motivated by the development in [3], which considers a single parabolic equation, we shall call a pair of functions (ii, D), both of which are in C’*‘(b x [0, T]) fl C’*‘(fi x (0, T]) a supersolution of the B.V.P. (2.1), (2.2) if ii and 0 satisfy the differential inequalities obtained by replacing “=” in the first equations of (2.1) and (2.2) by “2” and in the second equations of (2.1) and (2.2) by “5”. A subsolution of (2. l), (2.2) is defined by reversing the inequalities in the definition of a supersolution. By consideration of a quasimonotone system of parabolic equations equivalent to (2.1) (see [14, p. 581) and by straightforward modification of the


theory developed in [3] for a single equation, it follows that if (u, _u) is a subsolution of (2.1), (2.2), (ii, 0) is a supersolution, and u(x, 0) I ii(x, 0), _v(x, 0) 1 0(x, 0) for all x E si, then for any (&, &) E IE x IE such that for all x E fi

u(x, 0) 5 41(x) 5 W, 0), (2.4)

D(X, 0) 5 &(x) 5 _u(x, 0), (2.5)

there exists a unique solution (u, v) of the initial value problem (2.1) and (2.2) corresponding to the initial conditions U(X, 0) = r&(x), v(x, 0) = &(x) which is regular. That is, u and v are in C’90(sZ x [0, T]) n C2+a,1+a’2(fi x (0, ZJ). Moreover, l_dlUlii, D I v I y, and the Poincart map F from the subset of pairs (4i, $2) in IE x E satisfying (2.4) and (2.5) into E x E, defined by F(c#J,, q52) = (u( , T), v( , T)), is continuous and compact (cf. [3, proposition 5.11).

If (4r, 42) E P x P, M is a sufficiently large constant, u = 0, ii = M, _v = M, and 0 = 0, then (ii, 0) and (u, _v) are super- and subsolutions respectively of (2.1), (2.2), and (2.4), (2.5) hold. Therefore, F is defined on P x P, is compact and continuous, and maps order intervals in P x P into relatively compact subsets of P x P. That F is of class C’ on P x P follows from the fact that the system (2.1) is smooth in u and v and the obvious extension of the theory developed in [S, chapter III] to systems. It follows that condition (Al) is satisfied by F.

If (&, ti2), (~9,) 0,) E P x P, (&, c#I~) I (0,) 6,), and (u,, vl) and (u2, u2) are the solutions of the initial boundary value problems corresponding to the initial conditions (q& ,4~~) and (0,) d,), respectively, then it follows from the above discussion that

@I( 9 0, Vl( 9 0) 5 @2( 9 09 v,( 9 0) (2.6)

for I z 0, SO F(&, &) 5 F(B,, 0,). Suppose that & + 0, e2 f 0, 0 I & d et, 0 5 e2 5 $2, and #Jo f or. Since uk and uk are nonnegative for k = 1,2, it follows from (2.1) and (2.6) that there exists a constant y > 0, such that for x E Li and 0 I t I T

(u2 - u& - k,A(u2 - ul) + y(z.4, - ul) 2 0. (2.7)

If u,(.F, i) = U&F, f) for some (x, i) with K E Q and i > 0, then by the parabolic maximum principle [14], u1 = u2 for (x, t) E 51 x [0, f] contradicting $r f 6,. Therefore, ur < u2 on Sz x (0, T]). If u,(x, f) = 24,(x, f) for K E XJ and f> 0, then by (2.7) and the parabolic maximum principle

contradicting (2.2). Therefore, u1 < u2 on fi x (0, T]. It follows from (2.1) that for a sufficiently large positive constant y

(vi - v2)t - k2A(h - ~2) + Y(U, - ~2) 2 0 on d x (0, T],

>o somewhere in d x (0, T].

A repetition of the argument given above shows that v2 < ur on fi x (0, T]. Hence, F satisfies condition (A2).


If +r E PI and u is the solution of the I.V.B.P.

u, = ~,Au + u[a - bu],

au an = 0, w, 0) = 9h(x),

aoxlR+

(2.8)

then (u, 0) is the solution of the I.V.B.P. given by (2.1), (2.2) and the initial data U(X, 0) = &(x), u(x, 0) = 0. Therefore, iffr and fi are defined as in the previous section, then f2($1, 0) = 0 for all C#J~ E P and the same reasoning shows that fi(O, &) = 0 for all & E P.

In order to treat the problems

U, = k, Au + u[a - bu] 0nQxlR

(2.9)

u, = k2 Au + u[d - fv] 0nfixR

av

an = 0, u(x, t + T) = u(x, t) anxiR

(2.10)

we make use of the theory of principal eigenvalue of a periodic parabolic differential operator subject to Neumann boundary conditions. This concept was used in [5] and [l l] for the case of Dirichlet boundary conditions and in [6] for the case of Neumann boundary conditions. According to [6] there exist smooth functions cpi and (oz defined on fi x R and corresponding numbers CX~ and 01~ such that on fi x R

Pk(X, t + n = (P/&G 0, k= 1,2,

and vi and v2 are strictly positive on fi x I?.

The inhomogeneous periodic problem

ut - k, Au - au = g 0nQxR

au an = 0, u(x, t + T) = u(x, t),

aaxE

(2.11)

where g(x, t) is a T-periodic holder continuous function, satisfies the maximum principle [g > 0 * u > 0] if and only if c~i > 0. If a1 = 0 and g > 0, (2.11) has no solution at all.

PROPOSITION 2.1. There exists a positive solution u0 of (2.9) if and only if c~i < 0. The solution u0 is unique.

Similarly, there exists a (unique) positive solution u0 of (2.10) if and only if CY~ < 0.


Proof. The necessity follows from the fact that u0 > 0 violates the maximum principle

% - kl Au, - au, = -buz < 0.

For the sufficiency note that if 01~ < 0, then .zpl (E > 0 small) is a periodic subsolution, while a large constant A4 is a supersolution. Uniqueness follows from the concavity of the non- linearity just as in [9] (see remark added in proof) for the autonomous case.

Similar arguments hold for uo. Furthermore, one has the estimates

(2.12)

where subscript A4 (respectively L) denotes the maximum (minimum) of the function taken over d x R. (Of course the first set of inequalities is of interest only if a and d are positive on C? x R.)

If u,(x, 0) = C&(X) and v,(x, 0) = C&(X), then 6, and & are the unique fixed points in P - 10) of the mappings C#Q + f,(&, 0) and & + fi(O, &), respectively.

Suppose 0 < s < 1 and u is the solution of the initial value problem (2.8) where C#J~ = ~6,. If u = su,, then ZJ(X, 0) = u(x, 0), u satisfies the Neumann boundary condition, and

ut = k, Au + @[a - bu,] < k, Au + &[a - bu].

By an easy application of the parabolic maximum principle, it follows that for all x E 31, s&(x) = u(x, T) < u(x, T) and therefore, SC& Q fi(s&, 0). Since u(x, 0) = s&(x) < d,(x) = uo(x, 0) for all x E a, the same reasoning shows that f,(s&, 0) G &. A similar argument shows that C& + fi(s&, 0) Q SC& if s > 1. Since the corresponding result obviously holds for the mapping C#J~ + fi(O, &), we see that assumptions (A3) and (A4) of the previous section are satisfied.

To verify (A5), we note if uo, uo, $I, and & are as above, and h E P, then by the obvious extension of the theory developed in [B, chapter 31 to systems we have that D1 fi(O, &)h = z( , T) and D2 fi(&, 0)h = w( , T) where

zt = k, AZ + z[a - cuol, a.2 an

= 0, anx[o,Tl

(2.13)

W, = k,Aw + w[d - eu,],

aw - = an 0,

aQx[O,Tl

(2.14)

and z and w satisfy the initial conditions z(x, 0) = h(x), w(x, 0) = h(x). Since (2.13) has the trivial solution and z(x, 0) z 0, it follows from standard parabolic theory that z(x, t) I 0 on fi x [0, T], so zt - k, AZ + yz 2 0, where y is a sufficiently large positive constant. Therefore, if h 2 0, h + 0, it follows from the maximum principle that z(x, T) > 0. This shows that D, fi(O, $,)(P - {O)) c Int P and a similar argument shows that D, f2(&, O)(P - (0)) C Int P. Hence (A5) holds for F.

To obtain convenient expressions for the spectral radii of the linear maps Dlfi(O, 4,) and D, f2(&, , 0) we refer again to the principal eigenvalues. There exist smooth functions ‘//I and I,V~


defined on fi x [R and corresponding numbers pi and p2 such that on &? x R

vu - k Ay/, - Wl b - Chl = PI v/l 9 (2.15)

Wzt - k2AWz - W2[d - eu,l = P2v2

(2.16)

W/&G t + T) = WAX, 0, k = 1,2,

and vi and I,Y~ are strictly positive on fi x IR. We note that if 2(x, t) = e-plcyl(x, t), W(X, t) = e-PZ’y2(x, t), then z and w are solutions of (2.13) and (2.14), respectively, and if Ak = ee8k7, k = 1,2, then z(x, T) = A,z(x, 0), w(x, T) = & w(x, 0). By the above discussion, this means that if q(x) = z(x, 0) = vi(x, 0), v2(x) = w(x, 0) = wz(x, 0), then D,fi(O, &)u, = A,u,, D2 f2($i, 0)u2 = & v2. Therefore, by the Krein-Rutman theorem we have the following lemma.

LEMMA 2.2. If 131 and A2 are the spectral radii of Dlfi(O, 6,) and D2f2(rj1, 0), respectively, then A,> liffp,<OandAk< liffp,>Ofork= 1,2.

Let M,(t) = m~$u(x, 0 - c(x, 0v,(x, 01,

M,(t) = yy$4x, t) - 4x, tMx, 01.

PROPOSITION 2.3. If T

$‘ (a(~, t) - c(x, t)u,(x, t)) dx dt > 0,

0 a

then pi < 0 (hence Ai > 1). If

T

ilc (d(x, t) - e(x, t)u,(x, t)) dx dt > 0,

0 a

then & < 0 (hence A2 > 1). If

I

T

M,Jt) dt < 0, 0

then Pk > 0 (hence Ak < 1) for k = 1,2.

Proof. We write (2.15) in the form

Y1t -- k AWI -= [a - cvol + PI Wl l v/l

and integrate over the region Q x [0, T]. By 7’-periodicity of v/, , we have

(2.17)

(2.18)

(2.19)

T!h&&=O 9

0 v/l


and since r,~r satisfies the Neumann boundary conditions in x, it follows from Green’s first identity that

Consequently,

T

=- s i k 1 ‘Vy/,lZdwdt 5 0. 0 62x-

02 J 3 [a - cue] dtdx + &rlQ n 0

which shows that (2.17) implies PI < 0. The same argument shows that (2.18) implies p2 < 0. To prove the remaining part of the proposition, we consider the function

Since

P(f) = (! wl(x, 0 dt. a

integration of (2.15) with respect to the space variables over 51 gives

p’(t) = i

[4x, 0 - 4x9 O~o(X, OlWl(X, 0 dx + &p(t) n

for --03 < t < co. Since for each fixed t

.I [4x, t) - 4% t)uo(x, t)lvl(x, 0 dx 5 M,(t) u/1(x, t) dx = M,(t)p(t), 0 I 0

we obtain the differential inequality

p’(t) 5 M,(t)P(t) + &p(t).

Therefore, sincep(t) is positive for all t and T-periodic, by dividing byp(t) and integrating from 0 to T we obtain

i

T

05 M,(t) dt + /3r T. 0

A similar argument leads to the inequality

.i

T

01 Mz(t) dt + Pz T 0

and from these inequalities the second part of the proposition follows.


Remarks. 1. It is shown in [l] that if a&, . . . , f do not depend on x, then u,, and u,, are also functions of t alone. Therefore, in this case, the proposition is sharp in the sense that A, > 1 iff (2.17) holds, & > 1 iff (2.18) holds, and Ak < 1 iff (2.19) holds for k = 1,2.

2. It follows from propositions 2.3 and 2.1 that the positive solutions u0 and q, of (2.9) and (2.10) exist provided

T

IT

T

a(x, t) dx dt > 0, IS

d(x, t) dxdt > 0. (2.20) 0 n 0 D

We assume in the following that (2.20) is satisfied. The functions u. and u. cannot in general be found, but using the estimates (2.12) we can give

somewhat stronger conditions to ensure that Lk > 1 or Aik < 1 for k = 1,2.

PROPOSITION 2.4. If

then 2, > 1. If

/oT/o(a(x,O - (;)MC(X,f))dxdt>O,

.i:i,(d(x,t)- (;)$x,t))~dt>O,

then A, > 1. If d is positive and

(2.21)

(2.22)

s~yy;(@, 0 - ($;(x, 0) dt < 0,

then A, < 1. If a is positive and

(2.23)

(2.24)

then A, < 1. The proof is immediate from (2.12). From the last two results, theorem 1.1 and proposition 1.2, we have the following theorem.

THEOREM 2.5. Assume (2.20). If conditions (2.21) and (2.22) hold, then the mapping F associated with (2.1), (2.2) is compressive. More generally, if (2.17) and (2.18) hold, then F is compressive. If either (2.23) or (2.24) hold, then F is not compressive. More generally, F is not compressive if (2.19) holds for either k = 1 or k = 2.

Remark. 3. In [l] it was shown that F is compressive if

dM aL > --MT

fL

dL > ye, L

hold. Obviously the condition that both (2.21) and (2.22) hold is much weaker. Given a T-periodic solution (u, v) of the B.V.P. (2.1), (2.2), having both components

nonnegative with u(x, 0) = i,(x), v(x, 0) = 6,(x), we say that (u, u) is globally attracting if for every (c&, &) E (P - (0)) x (P - (O]), F@“(c$,, c&) --, (cfil, 4,) as m + 00. It is not difficult to show (see [l]) that if (u,, 0) is globally attracting, then for every solution (u, v) of (2.1), (2.2) with u(x, 0) 2 0, u(x, 0) L 0, u(x, 0) f 0, u(x, 0) + 0, it follows that u(x, t) - u,(x, t) -+ 0 and v(x, t) -+ 0 as t --+ 00 uniformly with respect to x E fi. The corresponding remark holds for

(0, uo).


In [l], it was shown that the conditions

imply that (u,, , 0) is globally attracting. The following result gives much weaker conditions. Let aL(t) = y2 a(x, t), b,(t) = F:; b(x, t), . . . .

THEOREM 2.6. Assume (2.20). If

(2.25)

(2.26)

then (u,, 0) is globally attracting. If the inequalities obtained from these by interchanging a and d, b and f, and c and e hold, then (0, vo) is globally attracting.

Proof. To prove the first part of the theorem we show that the conditions (2.25) and (2.26) imply that F has no coexistence state and A, > 1. The result then follows from theorem 1.5.

To this end assume that (2.25) and (2.26) hold and that, contrary to the above claim, there exists a coexistence state of F. This implies the existence of a solution (ii, 0) of (2. l), (2.2) which is T-periodic and has both components strictly positive on fi x R.

Let p(t), q(t) be defined by the differential equations

p’(t) = p(NaL(O - &&)N) - cdfMf)l

q’(t) = dN&At) - eL(OAt) - fLWdOl

and the initial conditions

It follows from the basic theory of such systems (see for example [2]) that p(t) > 0 and q(t) > 0 for all t 2 0. Viewed as functions of both x and t, u = p and v = q satisfy the system of inequalities

U, - k, Au I u[a - bu - cv],

v, - k2 Au 2 v[d - eu - fu],

u(x, 0) 5 22(x, O), u(x, 0) I 0(x, 0).

Therefore, by a basic comparison theorem for competition-diffussion systems [l], we have p(t) % ii(x, t), q(t) L 6(x, t) for all x E d and all t 2 0. In particular

p(T) 5 $; 12(x, T) = mEi: fi(x, 0) = p(0)

and q(T) 2 rJl;; ir(x, T) = y:; 6(x, 0) = q(0).


Therefore

(a~(0 - b,(tMO - c.dfMf)) dt

Osl;+dt=SoT (dM(t) - eL (Mt) - fL WqW dt.

If for a continuous function g(t) defined on [0, T] we let g denote the integral of g over this interval, then from the above we obtain

It follows from (2.25) and these inequalities that

hence (cM/fL)max(eL/bM)min < 1. On the other hand, (2.26) and the above inequalities imply I f

hence (CM/fL)max(eL/bM)min 2 1.

This contra&on shows’that no coexistence state exists if the conditions (2.25) and (2.26) hold.

To finish the proof, we show that (2.25) implies (2.17). Let r(t) be defined by the differential equation

r’(t) = dt)[dM(t) - fL(t)dt)l

and initial condition r(0) = max u,(x, 0). Viewed as both a function of x and t, r satisfies the differential inequality

xei=z

rt - k2 Ar L r[d(x, t) - f(x, t)r].

Therefore, since r(0) L u,(x, 0) for all x E a, r(t) I u,(x, t) for all 0 I t and x E a. In particular

so

r(T) 2 mEa; u,(x, T) = mEa; u,(x, 0) = r(O)

and T

c(x, t)uO(x, t) dxdt I lsZ[ fL (t)r(t) dt 0 I

Abstract competition model and applications

Therefore, assuming (2.25), we have

939

(a(x, t) - c(x, t)uO(x, t)) dx dt L lsZ1 [ /oTQO dt - cz)_? s:&&) df] > 0,

so (2.17) holds and 1, > 1. By earlier remarks, this proves the theorem.

To apply theorem 1.6, we assume a and d are positive on d x R and first verify (A6). If h E P, then (Difi(O, 0)/z)(x) = z(x, T), where z is the solution of the initial-boundary value problem

zt - k, AZ = az, az an =

0 anxR+

and z(x, 0) = h(x). It follows from the maximum principle that if h(x) 1 0, h(x) f 0, then z(x, t) > 0 for (x, t) E fi x (0, a). Hence, D,fi(O, 0) maps P - (0) into Int P. If h is an eigenvector of D,fi(O, 0) in P - (0) corresponding to the eigenvalue P and z is as above, then

z(x, T) = pz(x, 0) = ph(x) so

(P - 1) I

z(x, 0) du = ’ (z, - k, AZ) dxdt > 0. n 1.i 0 n

This shows that the spectral radius of D,fi(O, 0) is bigger than 1 and the exact same argument works for D,f,(O, 0). This establishes (A6).

From propositions 2.3 and 2.4 we have

THEOREM 2.7. If a and dare positive on d x R and (2.23) and (2.24) hold, then there exists an unstable coexistence state. More generally, this is true if (2.19) holds for k = 1, 2.

Acknowledgements-Research for this paper was initiated while the second author was a guest at the Mathematics Institute at the University of Zurich supported by the Swiss National Science Foundation. He is grateful for the invitation.

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