A limit set trichotomy for monotone nonlinear dynamical systems

Nonlinear Analysis, Theory, Methods&Applications, Vol. 19, No. 4, pp. 375-392, 1992

Printed in Great Britain.

0362-546X/92 IS.00 + 330 0 1992 Pergamon Press Ltd

A LIMIT SET TRICHOTOMY FOR MONOTONE NONLINEAR DYNAMICAL SYSTEMS

ULRICH KRAUSE and PETER RANFT

Fachbereich Mathematik/lnformatik, Universitgt Bremen, Bibliothekstrafie, 2800 Bremen 33, Germany

(Received 6 May 1991; receivedfor pubkution 4 October 1991)

Key words and phrases: Limit set trichotomy, monotone, discrete, nonlinear dynamical systems, part metric, cooperative differentiable systems.

1. INTRODUCTION

IN THIS paper we explore the conditions under which for nonlinear dynamical systems a limit set trichotomy is valid in the sense that for the (forward) orbits precisely one of the following cases does hold:

(i) each (nonzero) orbit is unbounded; (ii) each orbit converges to 0;

(iii) each (nonzero) orbit converges to a .unique positive fixed point. More suggestive, there is one center of attraction for all orbits which may be infinity, 0 or,

in between, some positive position. The natural settings for such a kind of asymptotic behavior are monotone (or, positive)

dynamical systems both in discrete as well as in continuous time. For those systems strong results have been obtained during the last years, including interesting applications to ecology, epidemiology, economics and demography (cf. the references given at the end of this paper, in particular the informative exposition in [13]). Concerning discrete time systems the asymptotic behavior of the iterates has been studied for operators leaving invariant the standard cone of the n-dimensional real vector space and which are assumed to exhibit certain properties of monotonicity and concavity (see Section 2 for the exact definitions). Recently, Smith [13] obtained nice results concerning those systems. Applying these results to the Poincare map of systems of ordinary differential equations Smith then was able to generalize important results by Hirsch [5] on cooperative differentiable systems.

Inspired by Smith’s paper we shall extend his results on discrete dynamics in various directions and apply it afterwards to cooperative differentiable systems. Our method of proof is rather different from that of Smith. Whereas Smith linearizes local nonlinearities and then uses common Perron-Frobenius theory on nonnegative matrices, we shall address nonlinearities directly by using some nonlinear type of Perron-Frobenius theory along the lines developed in [8]. (Although we shall stick to finite dimensions the methods as developed in [8] apply to infinite dimensions.) This enables us to avoid any differentiability assumption in proving the limit set trichotomy for discrete systems. Furthermore, by working piecewise on the so called parts of a convex cone and employing the respective part metric (see Section 2 for the definition) it is possible to weaken the assumptions of concavity. As in the case of Smith’s paper our paper too is related to the pioneering, albeit not yet fully acknowledged, work of Krasnosel’skii [7] (cf. also [S] on the relationship to the work by Krasnosel’skii and his collaborators).

375

376 U. KRAUSE and P. RANFT

In the next section of the present paper the necessary notations and definitions will first be given. In particular, the (k, P)-property will be defined by which we generalize Smith’s concavity condition which in turn generalizes an essential condition employed by Hirsch. Theorem 1 presents the main result of this section saying that for continuous and monotone operators possessing the above property the limit set trichotomy is valid on parts. By special- izing theorem 1 and casting the (k, P)-property in terms of differentiability we obtain with theorem 2 a result which is close in spirit to Smith’s key result on discrete dynamics but still more general than this. In the last section we shall apply these theorems to get via the Poincart map with theorems 3 and 4 a limit set trichotomy for differentiable systems. Thereafter we discuss applications to models from ecology and epidemiology.

2. LIMIT SET TRICHOTOMY FOR DISCRETE NONLINEAR SYSTEMS

Let R be the set of real numbers and let R” = lx = (x1 , . . . , x,) 1 Xi E R for all il. A convex cone in R” is a nonempty set C c L?" having the following two properties:

(a) x, Y E C implies x + Y E C (b) x E C and i E R, 2 > 0, imply Ax E C.

Particularly interesting convex cones in what follows are the standard cone

IR: = {x E I?” ) xi 2 0 for all i)

and its parts. A part of I??+ thereby is defined as an equivalence class with respect to the equivalence relation

X-Y iff for some h > 0 ix5y+x.

Here I denotes the ordering relation induced by the standard cone, i.e. x I Y iff Xi I Yi for all i E_n = (1, . ..) n]. It is easily verified that parts P are convex cones which correspond in a unique manner to the subsets Z C _n by

P=(x~lR~(x~>Ofori~Zandx~=Oforj$Z).

For example, Z = Qr corresponds to P = (0) and Z = _n corresponds to P = int IR:, the interior of the standard cone (for the Euclidean topology).

For a subset Z c _n and x, y E lR” we write x <r y iff x I y and Xi < _Yi for all i E I, e.g. x <+ y means x 5 y and x <n y means xi < Yi for all i E _n which we denote shortly by x < y. If x 5 y but x # y we shall use the notation x 3 y. By viewing matrices as vectors the above relations also apply in an obvious manner to matrices.

Consider a proper part P of RT , that is Z # 0 for the corresponding index set. Let for

X,YEP

n(x,y) = sup(A rOIAx5y) = min c isZ . II I I

By setting p(X, y) = -log min(A(x, y), A(Y, X)) = maX( IlOg Yi - log Xi1 1 i’E 11

a metric on P is defined which is known as the part metric (cf. [3, 141). Let X be some nonempty compact set (for the Euclidean topology) which is contained in

some part P with index set I. Then on X the part metric is equivalent to the maximum metric 11x - y/J = max( [xi - yil 1 i E _n). More precisely, because for any two positive real numbers r, s

Limit set trichotomy 311

there exists a number t between the two such that Ir - SI = tllog r - logsl, it follows that

W(X,Y) 5 IIX - Yll 5 M&Y)

where a = inf(z, ) z E X, i E I) > 0 and b = sup(zi I z E X, i E I) < 00. After having introduced the necessary notions as far as convex cones are considered, we now

turn to operators leaving the standard cone invariant. An operator T: I?: --) I?: is monotone, ifOIx<yimpliesTxI Ty.ForkEN =(1,2,3 , . . .I and P a proper part of IR: with index set I, an operator T: I?: --t IR: is said to have the (k, P)-property, if for all 0 < ,I c 1 and all 0 cI x it holds that

ATkx cr Tk(Ax).

(Tk denotes the kth iterate of T.)

Remark. For k = 1, P = int R: the above property corresponds to Krasnosel’skii’s strong concavity [7]. An operator may possess the above property for some k and P without possessing it for k = 1 or for P = int I?:. Later on, after having derived criteria for operators to possess the above property, this property will be discussed further, in particular its relationship to Smith’s notions of monotonicity and concavity will be examined.

Now we can state and prove our limit set trichotomy for orbits generated by an operator T, i.e. for the sets O(x) = (T”‘x 1 m 2 0) with x E I?“, .

THEOREM 1. Let T: I?: -+ I?: be a monotone and continuous (for the Euclidean topology) operator with the (k, P)-property for some k and some proper part P. If P contains an orbit at all then precisely one of the following three cases does hold:

(i) each nonzero orbit in P is unbounded; (ii) each orbit in P is bounded with at least one limit point not contained in P;

(iii) each nonzero orbit in P converges to the unique fixed point of T in P.

Proof. (1) First we show that if one orbit in P is bounded or, if the closure of one orbit is contained in P, then the same is true for all orbits in P. Let O(x), O(y) c P and suppose O(y) is bounded, hence T”‘y I z for some z E IR: and all m L 0. There exists 0 < A < 1 such that Ax I Y. The (k, P)-property together with monotonicity implies especially ATkx I Tky. Since 0 <I Tkx, Z the index set for P, it follows by iteration that ATmkx 5 Tmky for all m L 0. Hence Tmkx I (l/,I)z for all m. By monotonicity of T

Tmk+jxS T-+)s~~~Ti(~z) forallm,alljE[O,...,k- 1).

Thus O(x) is also bounded. This shows, that if one orbit in P is bounded, all orbits in P must be bounded. Let O(x), O(y) c P and suppose now that the closure cl O(y) (for the Euclidean topology) is contained in P. There exists 0 < ,I < 1 such that hy I T’x for all j E (0, . . . , k - 1). As before, this implies lTmky 5 Tmk’jx for all m 2 0 and j E (0, . . . , k - 1). Since cl O(y) C P = (z E I?: 1 zi > 0 for i E I, Zj = 0 for j $ I) there exists some r > 0 such that (Tmy)i 2 r > 0 for all m 2 0, all i E 1. It follows that (Tmk+jX)i 2 r for all m 1 0, j E (0, . . . . k - 1) i E 1. Since O(x) C P, (TmX)i = 0 for all m 1 0, all i $ I. Thus we must have cl O(x) C P. This shows, that if the closure of one orbit is contained in P, the closure of each orbit in P must also be contained in P.


(2) Suppose now, neither case (i) nor case (ii) does hold. Then P must contain a bounded orbit. By step (1) all orbits in P must be bounded. Since (ii) does not hold by assumption, there is at least one bounded orbit in P, the limit points of which all belong to P. By step (1) for all orbits in P the closure is also contained in P. Let 0(x,) be an orbit in P. We shall show that this orbit converges to the unique fixed point of T in P. By the above, the set X = cl 0(x,) is bounded and hence compact for the Euclidean topology. Furthermore, X c P. We have already seen, that part metric p and maximum metric are equivalent on compact subsets of a part. Therefore (X, p) is a nonempty compact metric space. Since T is continuous, T: X -+ X.

We shall show, that Tk is a contraction on the metric space (X, p). If x, y E X and 0 < A < 1 such that Ax I y, then the (k, P)-property together with monotonicity for T imply that (Tky)i - A(Tk~)i > 0 for all i E I. Tkx E X C P implies (Tkx); > 0 for all i E I. Consider for O<A<l

- - Since T is continuous on the compact space X there exist x,y E X and i, E I such that

Choose x,y E X with x # y. For J. = min(A(x,y), A(y, x)) 0 < A < 1 and Ax I y, Ay I x. Because of (Tk~)i = (Tky)i = 0 for i $ Z from the definition of 9 it follows that &A)Tkx I T”y and p(A)Tky I Tkx. Especially, Tkx and Tky belong to the same part. Putting

,D = min(n(Tkx, Tky), A(Tky, Tkx)),

it follows that (D(A) I ,u. Hence

p(Tkx, Tky) = -1ogp 5 -log p(n) < -log A = p(x, y).

Thus Tk is a contraction on the compact metric space (X, p). (3) A well-known variant of Banach’s contraction mapping principle applied to the contrac-

tion Tk on (X, p) yields, for every x E X, the convergence of the sequence (Tmkx), with respect to the part metric to a uniquely determined fixed point 4 of Tk in X. Since for x E X also T’x E X, it follows the convergence of O(x) with respect to the part metric to q for each x E X. It follows that even Tq = q because T is continuous on X for the Euclidean topology and a fortiori for the part metric. Thus lim Tmxo = q E P, and the limit applies also with respect

m-m to Euclidean topology by the equivalence of part metric and maximum metric on X.

It remains to show that q, which may still depend on x0, is the only fixed point of T in P. Suppose q’ # q is another fixed point of T in P. A4 = (q, q’) is a compact set contained in P. The same procedure as in step (2) when applied to M instead of X yields that Tk is a contraction on M. Therefore p(q, q’) = p(Tkq, Tkq’) < p(q, q’), which is a contradiction. This proves the theorem. w

Remark. Theorem 1 may also be stated as follows: Let T: lR= -+ F?z be a monotone and continuous operator with the (k, P)-property and such that the proper part P contains a bounded orbit and an orbit together with its limit points. Then each orbit in P must converge to the unique fixed point of T in P.


For the particular case where the part is the interior of the standard cone, theorem 1 yields the following consequence.

COROLLARY 1. Let T: Ii?: + F?: be a monotone and continuous operator having the following two properties:

(a) there exists some k such that for all 0 < A < 1 and all 0 < x

AT”x < Tk(Ax);

(b) there exists some 1 such that T’x > 0 for all x ? 0. Then precisely one of the following three cases does hold:

(i) each nonzero orbit is unbounded; (ii) each orbit converges to 0, the unique fixed point of T;

(iii) each nonzero orbit converges to q > 0, the unique nonzero fixed point of T.

Proof. (1) First we show that Tmx : 0 for all x ‘+ 0 and m 2 0. Suppose, on the contrary, T”x = 0 for some x 5 0 and m z 0. Then TmO 5 T”‘x = 0 by monotonicity and hence T”‘x = T”0 = 0. It follows that TZmx = T”(Tmx) = T”O = 0 and Tlmx = 0 by iteration. But T’x > 0 by assumption and hence by iteration T”“x > 0. Therefore we must have T”x ? 0 for all x 5 0 and all m 2 0. From this it follows that T ‘+m~ = T’(Tmx) > 0 for all x 5 0 and all m 5: 0. This shows that all orbits with the possible exception of the zero-orbit stay finally in int R”, .

(2) We apply theorem 1 for P = int RT. By step (1) cases (i) and (iii) in the trichotomy of theorem 1 become cases (i) and (iii) as stated in the corollary. Concerning case (ii) suppose there is an orbit O(x) having no subsequence converging to 0. Then there exist i E _n, r > 0 and a natural number m, such that ( Tmx)i 2 r for all m 2 m, . Define e E iR= by ei = r and ej = 0 for j # i. Then we have Tmx L e and T”(T’x) = T’Tmx 2 T’e by monotonicity. Since T’e > 0, it follows that the orbit starting in T’x has all its limit points in P = int IR”, . But this contradicts case (ii) of theorem 1. Hence every orbit has a subsequence converging to 0. Consider an orbit O(x) for x > 0. Then in particular Tdx I x for some d E N, and the Tmdx form a decreasing sequence in m which must converge to some y E II?: for m -+ co. By continuity of Td, Tdy = y. On the other hand, the orbit O(y) must also have a subsequence converging to 0. Therefore, Td’y = 0 for some d’ and y = 0 by step (1). This shows that li_m, Tmdx = 0 (in the Euclidean

topology). If x $ 0, then T’x : 0 for all r 2 0 by step (1) and hence T’+‘x > 0. Therefore lim Tmd+/+r

x = 0 for all x $ 0, for all r > 0, i.e. O(x) converges to 0. This proves the m-m corollary. n

Remarks. (1) If in the above corollary T satisfies in addition TO 2 0 then only the cases (i) and (iii) survive because of T”0 L TO $ 0. Thus if T has a fixed point and TO s 0 then case (iii) must necessarily hold.

(2) One may ask if something more can be said in case (i) in the sense of converging quotients of components of the orbit. This actually is true as may be seen from nonlinear Perron- Frobenius theory. If (1. I( is the Euclidean norm on R” and U = lx E W: 1 [Ix/I = 1) then under the assumptions of corollary 1 the operator T is 1) * )I- ascending in the sense (cf. [8]) that for every 0 < A < 1 and every x, y E U the inequality Ax I y implies p(A)Tkx I Tky. There V: [0, l] + [0, 11 is a continuous mapping with p(A) > A for 0 < A < 1 which by the compact- ness of U is obtained essentially in the same manner as the function v, in step (2) of the proof


of theorem 1. T being 11. II- ascending the nonlinear eigenvalue equation TX = Ax has a unique solution x* E U, I* > 0 and the orbit with respect to the resealed operator TX = Tx//Txll converges for each x 2 0 to x* [8, theorem 31. Convergence of the f-orbits expresses convergence of the proportions of the T-orbit.

The crucial condition in theorem 1 as well as in corollary 1 is the (k, P)-property for the operator T. Hence some criteria to check that (k, P)-property for operators will be useful. The following criteria employ the Jacobian DT(x) of T in x.

LEMMA 1. For an operator T: R: + I?“, and a nonempty index set Z C _n defining the part P, the following criteria are available.

(1) If DT(x) exists for all 0 <I x then T has the (1, P)-property iff DT(x)x cr TX holds for all 0 cl x.

(2) Suppose for some natural number k and every 0 <I x the matrix product

M(x) = DT(Tk-‘x) - DT(Tk-‘x) ... DT(Tx) . DT(x)

does exist. If for 0 <, x and 0 < r < 1 given M(x) I M(tx) holds and if for every i E I there exists at least onej E Z such that M,(x) < Mij(tX), then T has the (k, P)-property.

(3) T has the (2, P)-property if the following two conditions are satisfied: (a) 0 <1x 4 y implies 0 <1 TX and DT(x), DT(y) exist with 0 I DT(y) d DT(x); (b) for every 0 <, x, every 0 < t < 1 and every h E I there exist indices i E _n, j E Z such

that (DT(Tx)),, > 0 and (DT(X))ij < (DT(tX)),.

Proof. (1) Consider for i E _n and 0 <, x the function fi,, defined by f;,,(A) = (T (Ax))/A for A > 0; thereby T denotes the ith component mapping of T. If T has the (1, P)-property, then for arbitrary 0 < A < p there holds (A/,u)T(px) <, T@x), i.e. (T(px))/p <r (T(;lx))/A. Thus& is decreasing for all i E _n and strictly decreasing for all i E I. Conversely, supposefj,x possesses these properties. It follows that for 0 < A < 1, h,,(l) I f,,,(A) for i E _n and A,,(l) <f,,,(A) for i E I. Hence TX cl (T(Ax))/A. This shows that the (1, P)-property for T is equivalent to the above decreasing behavior of the functions f;,, . By assumption, fi,, is differentiable with respect to A and the chain rule gives

fi’,xtk) =

ADT(Ax)x - T((nx)

A2 .

Thus the (1, P)-property for T is equivalent to the property that for all y = Ax, 0 < A, 0 <( x, DT(y)y - 7;(y) 5 0 for i ~_n and DT(y)y - T(y) < 0 for i E I. That is, the (1, P)-property for T is equivalent to DT(y)y <I Ty for all 0 <, y.

(2) By the chain rule

DTk(x) = DT(Tk-‘x) . DT(Tk-2~) . . . DT(Tx) * DT(x).

The mean value theorem for the ith component mapping Tk of the iterate Tk yields

I-?x = DT+(tix)x + qkO with 0 < ti < 1.


By assumption, the ith row M;(x) is smaller than the ith row Mi(tiX) with at least one component with index in Z strictly smaller provided i E 1. Hence for i E Z

Mi(X)X < Mi(tiX)X = o~;.k(tiX)X I TikX . X

for all 0 <[x because of TkO 2 0. Similarly, for i @I Mi(X)X % Tkx - x. Therefore DTk(x)x -cI Tkx - x for all 0 <I x. By criterion (1) Tk has the (1, p)-property, that is, T has the (k, P)-property.

(3) Let 0 <I x and 0 < t < 1 be given. By the mean value theorem for every i E g

TX = D7;(tiX)(l - t)X + T(tX)

with t < ti < 1. From condition (a) it follows that 6x 2 T(tx) for all i, i.e. T(tx) I T(x). Since 0 <r T(tx), (a) implies 0 I DT(Tx) % DT(T(tx)). Also by (a), 0 I DT(x) 5 DT(tx). Hence M(x) = DT(Tx) - DT(x) exists and M(x) I M(tx). Furthermore, by condition (b) for h E Z there exists j E Z such that

Mhj(X) = C (DT(TX)),i(DT(X))ij < C (oT(T(iX))),i(oT(tX))ij = Mhj(tX)* i i

By criterion (2) T has the (2, P)-property. n

Remark. Considering the case Z = g and using a different argument it has been shown in [7, lemma 10.31 that DT(x)x < TX for all 0 < x implies strong concavity ((1, int I?:)-property) for T.

Using lemma 1 in connection with corollary 1 we obtain the following theorem which is useful in applications and which generalizes a theorem of Smith [ 13, theorem 2.11.

THEOREM 2. Let T: IR: + F?: be a continuous operator mapping the interior of IR: into itself and differentiable on the interior of R: . Assume the following conditions:

(a) 0 < x I y implies 0 I LIT(y) 5 LIT(x);

(b) there exists a natural number m such that for every 0 < x and every 0 < t < 1

0 < DT”(x) 2 DT”(tx).

Then the following trichotomy holds: (i) each nonzero orbit is unbounded;

or (ii) each orbit converges to 0, the unique fixed point of T;

or (iii) each nonzero orbit converges to q > 0, the unique nonzero fixed point of T.

Proof. From (a) it follows by the mean value theorem that T is monotone on int IR: . Continuity of T implies monotonicity of T on IR:. Since T maps int IRY into itself and is differentiable on int R: it follows that LIT”(x) = DT(T”-‘x) -.a DT( Tx)DT(x) exists for all 0 < x. From (a) it follows by the monotonicity of T that 0 < x I y implies 0 I DTm(y) I

DTm(x). Let P = int IR: with Z = _n and S = Tm. We want to apply criterion (3) of lemma 1 to S and P. By the above we have that 0 <I x I y implies 0 <I Sx and 0 I DS(y) 5 DS(x). Furthermore, from assumption (b) we obtain, for 0 < x and 0 < t < 1 given, indices i, j E Z

such that (DS(X))ij < (DS(tX))ij and (DS(SX)),i > 0 for every h E I. By criterion (3) of lemma 1

382 LJ. KRAUSE and P. RANFT

S therefore has the (2, P)-property, i.e. T has the (2~7, P)-property for P = int I?:. To apply corollary 1 we shall show that T”x > 0 for all x < 0. For x $ 0 given there exists 0 < Y with x 5 Y. For 0 < E < 1, i E _n, by the mean value theorem there exists 0 < t < 1 such that

Si(X + &_Y) = DLSi(t(x + Ey))(X + &_Y) + ISi(

Since 0 < t(x -t &y) I: 2y it follows that

S;(X + &y) L DSi(2y)(X + &y)*

For E converging to 0 the continuity of Si yields Si(X) 2 DSi(2Y)X. Since DSi(2Y) > 0 by assumption (b) and x : 0 it follows that Si(X) > 0. Thus T”x = S(x) > 0 for x $ 0. Applying corollary 1 then yields the wanted trichotomy. n

We conclude this section by discussing the relationship of our theorems 1 and 2 as well as their relationship to a theorem by Smith [13, theorem 2.11 which is similar in spirit. Smith assumes the following kinds of monotonicity (M) and concavity (C), respectively

DT(x) > 0 ifx>O (M)

DT(Y) 5 DT(x) if 0 < x < y. (C)

(M) implies that T maps int RT into itself and is part of our assumption (b) with m = 1. The remaining part of (b) for m = 1 as well as assumption (a) are implied by (C). Beside (M), (C), continuity and continuous differentiability of Ton int R”, , Smith also assumes the existence of LIT(O) with lim,,,,, LIT(x) = DT(0). Therefore in theorem 2 the assumptions are weaker than those in the theorem of Smith (by his assumptions, however, Smith is able to characterize for TO = 0 the convergence to 0 by the property that the Perron-Frobenius eigenvalue of LIT(O) does not exceed the value 1). A simple example which is covered by theorem 2 but not by Smith’s theorem is given by TX = (fi + 3x,, 2x,) for x = (x1, x2) E R”, . Neither does LIT(O) exist nor is (M) satisfied for that example. The assumptions of theorem 2, however, are satisfied. Thereby assumption (b) is satisfied for m = 2 but not for m = 1. In this example T

has the (2, P)-property but not the (1, P)-property. By theorem 2 the trichotomy holds for T.

This trichotomy together with the observations that 0 is the only fixed point of T and that T2”‘x 2 x for all m r 0, all x 2 0 yields that in this example all orbits with starting point different from 0 are unbounded, as indicated in Fig. Al (hence the general remark (2) made above after corollary 1 may be applied to analyse that particular example further).

An even more simple example which is neither covered by Smith’s theorem nor by theorem 2, due to lack of “cooperation” among the coordinates, is given by TX = (4x,, G) for x = (x, , x2) E R:. Corollary 1 is also not applicable because there is no I such that T’x > 0 for all x 2 0. The operator T, however, possesses for all proper parts the (1, P)-property. This can be easily seen directly or by criterion (1) of lemma 1. Therefore, theorem 1 may be applied to each proper part separately to yield trichotomy on it. Because each of the possible parts PI = ((0, O)), P2 = int R, X (OJ, P3 = (01 X int R,, P4 = int LR: contains precisely one fixed point, namely q1 = (0, 0), q2 = (1, 0), q3 = (0, l), q4 = (1, l), respectively, the cases (i) and (ii) in the trichotomy are not possible. Hence, on each part all the orbits converge to the unique fixed point of that part as shown in Fig. A2.


A case like this cannot be handled by theorem 2 or by Smith’s theorem, because according to these theorems the operator T has at most two different fixed points. Thus, of the theorems discussed so far, theorem 1 is the most powerful one, although sometimes not so easy to apply.

3. LIMIT SET TRICHOTOMY FOR COOPERATIVE DIFFERENTIABLE SYSTEMS

As an application of the above theory we consider differential equations which are given on I?:, the nonnegative cone of R”. Such systems appear frequently in mathematical models of ecology, epidemiology and economy where IR: is the natural state space. We consider the following system

x’ = F(x, t) xERT,tElR+ (*)

with F = (F,,F,, . . . , F,) r-periodic in r E R, for some T > 0, continuous in (x, t) E IR: x IR, and continuously differentiable in x E int Rt.

We assume that solutions ~(t, x) of (*) with initial condition ~(0, x) = x E IF?“, exist and are unique with respect to x and are defined for all t 2 0 (for existence and uniqueness it suffices that F is locally Lipschitz in x and a CarathCodory function).

We shall employ the following assumptions for the above system.

Weak positivity (w). For all i E n let 4(x, t) > 0 when x 2 0 and Xi = 0.

Cooperativity and irreducibility (ci)

2 (x, t) 1 0 for i # j, x E int I?“, , t 1 0, i, j E _n (cooperativity) J

D,F(x, t) = [ I

2 (x9 t) is irreducible for x E int IRT , t L 0. J ij

A matrix A E lRnxn is called irreducible iff no nonzero proper subspace spanned by a subset of the standard basis of IR” is mapped by A into itself. We say for short that F is cooperative (resp. irreducible) if the Jacobian matrix satisfies the corresponding conditions in (ci) above.

The following lemmata are needed in the proof of theorems 3 and 4. Lemma 3 is a useful comparison theorem of Kamke [6] and Miiller [l 11.

LEMMA 2. Consider the above system (*) x’ = F(x, t). Then: (i) F is weakly positive * p(t, IR”,) C lR= for t 1 0;

(ii) F is cooperative and irreducible * D,p(t, x) > 0 for (t, x) E IR, x int I?;.

For a proof see [l, p. 115, lemma 21 and [5, p. 426, theorem 1.11.

LEMMA 3. Let cy and 0 be solutions of x’ = g(x, t) and x’ = h(x, t) respectively. Where v/, 0 are defined for [to, tJ and are unique w.r.t. initial conditions &to, x0) = x,,, S(t,, y,,) = y,. Assume that g or h is cooperative, g I h and x0 < y,. Then it follows that cp(t, x0) < O(t, y,,) for t E PO, t11.

For a proof see [6; 11; 4, p. 28, theorem 8; 10, p. 30, theorem 1.7.31.


For (*) consider the Poincare’ map T given by T: IR: -+ R:, TX := ~(r, x). If F is time independent we define TX := ~(1, x). There is an obvious correspondence between fixed points of T and periodic solutions of (*). If F is cooperative and irreducible it follows from lemma 2(ii) that for 0 < x < y we have

c

1

cp(l,Y) - v(t, x) = D*p(t,x + S' (y - x)) * (y - x)d5 > 0, 0

i.e.(o(t,x)<~(t,y)forallO<x<y,trO.Hencefort=zandO<x<ywehaveTx<Ty and by continuity of T we get TX I Ty for all 0 s x I y. Thus we are led to a situation with a discrete nonlinear operator on a convex cone which is monotone.

In order to apply theorem 1 (resp. theorem 2) to the Poincare map T we employ two concavity conditions for the right-hand side of a cooperative and irreducible system (*). The proper part in consideration is P = int m:.

Ray concavity (rc). F is called ray concave if D,F(x, t) 2 D,F(cY * x, t) for x > 0, 0 < CY < 1 and t 2 0.

Weak concavity (WC). F is called weakly concave if F is ray concave and D,F(y, t) I D,F(x, t) for 0 < x I y,

t 2 0.

The first property (rc) leads to theorem 3 which will be proved by applying theorem 1 of Section 2 to the Poincare map of (*). Theorem 3 says that trajectories of cooperative, irreducible, ray concave systems tend either to infinity (i) or to the boundary of E?T (ii) or to a periodic trajectory in the interior of IR: (iii). Although that is not the full limit set trichotomy (referring to case (ii)), we get in (ii) a reduction of the dimension and know that all trajectories are bounded. (As has been pointed out to the authors by H. R. Thieme, tending to infinity in the sense of (i) does not automatically imply convergence to 00 in all components.)

Below we give an example which is ray concave but not weakly concave. It can be shown that for a solution cp(t, x) of a ray concave system the interval of existence is IR, (cf. Appendix).

Assuming that F is in addition weakly concave we show.with theorem 4 that there is a limit set trichotomy in the sense that all trajectories either tend to infinity or to zero or to a periodic trajectory in the interior of K?;.

THEOREM 3. For (*) x’ = F(x, t) as above assume (w), (ci) and (rc). Then exactly one of the following situations occur:

(i) p(k * T, x) is unbounded for all x E int IR: ; (ii) rp(k - T, x) is bounded and at least one limit point of cp(k * t, x) is contained in ~3lR: for

all x E int IRT; (iii) &k - 5, x) z q > 0 for all x E int R:, t&t, q) is r-periodic.

Proof. Let F be cooperative, irreducible and ray concave. We know that T: IR’J -+ I?:, TX = ~(7, x) is continuous and monotone. In addition T is continuous differentiable in int ll?: . We show that T has the (1, int lR:)-property by applying criterion (1) of lemma 1. Theorem 3 follows then from theorem 1.


CLAIM. T has the (1, int R:)-property, that is: ATx c T(Ax) for all x > 0, 0 < I < 1.

Proof of the claim. By lemma 1 criterion (1) the above claim is equivalent to:

DT(x)*x < TX for all x > 0.

Let x > 0, t 2 0 and define q(t) := p(t, x) - D,cp(t, x) * x. We show that q is positive. We have

q’(t) = F(V(f, 4, 0 - R&P(t, x), 0 * ac4t, xl * x

+ DJ?W, x), 0 * Co@, 4 - wTv(t, -9, 0 * df, x)

= Qmdt, xl, 0 * do + HO

with b(t) := F(q(t, x), t) - D,F(p(t, x), t) * cp(t, x). By the variation of constants formula we get

q(t) =

i

f

v(t, xl * V’(& 4 * w ds

0

where ty(t, x) * I,-‘(s, x) is the fundamental matrix of x’ = D,F(yl(t, x), t) * x at time t = s and hence v(t, x) * t,v-‘(s, x) > 0 for t > s (compare [1, lemma 21). Let y > 0, t 2 0 and i E _n. Then by the mean value theorem there exist 0 < oli < 1 such that

F,(Y, t, - 4(O, t, = DXC(aiY, t, ‘Y

and since F is ray concave it follows that F(Y, t) - D,F(y, t) *y : 0 for all y > 0, t 2 0. We know that p(t, x) > 0 thus b(t) ? 0 for t 2 0 and hence q(t) = &t, x) - III&t, x) * x > 0. For t = T it follows that TX - LIT(x) * x > 0 for all x > 0. n

Example 1. The following system satisfies (rc) but not (WC). Moreover the y-axis is invariant and the right-hand side is not differentiable at (0,O).

X’

Y'

where (x, y) E IR: and let F : = (Fl , is given by

= (x.y)“4 - x = : F,(x, y)

= x - y =: F,(x,y)

F,). F is weakly positive and the Jacobian of F at (x, y) > 0

r (x * yy4 * (x - yy4 1 ___- ___ D,F(x, Y) =

1

4 * x 4-y

1 -1 1

thus F is cooperative and irreducible. F is not weakly concave since (1, 1) < (2, 16) and

$(l, 1) = ; - 1 < 7 - 1 = $32, 16).

Let (x, y) > 0, 0 < CY < 1 then

G>l (x * y)“4 lG (x*y)“4

=$ ~(cY.x,cX.y)>~(x,y) CY * 7.;’ 4-Y ay aY


and similarly for Z,/CIx. This shows that F is ray concave. The equilibrium points are (0,O) and (1, 1). We can apply theorem 3 and get ~(t, (x, y)) -+ (1, 1) as t -+ ~0, for x 2 0, y > 0 and &t, (0,~)) -+ 0 as t -+ 00, for y I 0 (compare Fig. A3).

We come now to the case where F satisfies (WC).

THEOREM 4. For (*) x’ = F(x, t) as above assume (w), (ci) and (WC). Then exactly one of the following situations occurs:

(i) &k * t, x) is unbounded for all x E RT\{O); (ii) p(k . r, x) k-, 0 for all x E Rt\(O];

(iii) fj7(k - 5, x) k-,’ q > 0 for all x E R:\(O), p(t, q) is r-periodic;

i.e. we have a limit set trichotomy.

Proof. Let F be weakly positive, cooperative, irreducible and weakly concave. As above T: R: -+ I?;, TX = cp(z, x) is the Poincare map of (*). We show that T satisfies (a) and (b) of theorem 2 (for m = 1). Theorem 4 is then a direct consequence of theorem 2.

We have to show (a) 0 < x 5 y * DT(y) s DT(x)

(b) O<a< l,x>O * O<DT(X)~DT(CY~X).

We prefer this kind of proof rather than applying theorem 3 directly and repeating in case (ii) the proof of theorem 2.

Proof (a). Let 0 < x 5 y. It follows that cp(t, x) I cp(t, y) for t L 0 and by assumption we have that D,F(cp(t, y), t) 5 D,F(cp(t, x), t). Write D(t, x) := D,q$t, x) for short. D is a solution of the variational equation D’ = D,F(q(t, x), t) * D. With the above inequality and lemma 3 it follows that D(t,y) I D(t, x), t 2 0 and in particular for t = t that DT(y) I DT(x) for o<x<y.

Proof(b). Let x > 0, 0 < EY < 1. From lemma 2 we know that D,p(r, x) = DT(x) > 0. We write as above D(t, x) := D,p(t, x) and M(t) := D(t, a * x) - D(t, x). It follows that

M’ = D,F(p(t, a * x), t) * D(t, CY * x) - D,F(p(t, x), t) . D(t, x)

= D,F(p(t, a . x), t) . A4 + [D,F(&, a * x), t) - D,F(p(t, x), t)] . D(t, A-).

Variation of constants yields

M(t) = D(t, a * x) . D-‘(s, a * x) * [D,F(&s, a * x), s) - D,F(rp(s, x), s)] * D(s, x) d.s

D(t, CY * x) * D-l@, CY - x) is the fundamental matrix at time t = s of the above variational equation, i.e. this matrix is positive for t > s (compare [l, lemma 21).

That is D(t, a * x) - D-‘(s, a * x) > 0 for t > s and D(s, x) > 0. For s = 0 the assumptions imply that D,F(a * x, s) - D,F(x, s) s 0. Since F is continuously differentiable in int R”, , it follows from the continuity of the derivative that M(t) z 0, i.e. DT(x) 2 DT(a * x) for t = 7. n


Example 2. In the following autonomous system the differentiability condition of [13] is not satisfied at (0, 0).

x’ = -_x + I 2*y+*‘v5=:F*(x,y)

y’ = -y + $ . x + + * fi =: F,(x, y)

(x,Y) E R2,,F:= (F,,F,). F is weakly positive and continuously differentiable in int RT. The Jacobian of F at

(x,y) > 0 is given by

and hence F is cooperative and irreducible. The equilibrium points are (0,O) and (1, 1). The assumptions of theorem 4 are satisfied and it follows that ~(t, R:\lO]) r+m (1, 1) (compare Fig. A4).

Remark. The above theorems and the proofs are closely related to ideas in articles of Smith [13], Hirsch [5] and Krasnosel’skii [7]. The main difference is the weaker concavity condition in theorems 3 and 4 and that we do not need a differentiability condition at 0. The corresponding concavity condition (C) of [13] is D,F(y, t) I D,F(x, t) for 0 < x < y, t 2 0.

For biological models it is quite natural that there is an invariant subset of the boundary iR= since if one,species dies out it cannot recover again. Theorem 3 is applicable to systems which leave a subset of 8lR: invariant (see example 1). Since we have TX > 0 for x s 0 this cannot happen for weakly concave systems and theorem 4 cannot be applied.

In the following we illustrate the importance of the above class of differential equations in certain applications. In some cases we can deal with systems which are not yet covered in the literature.

Positive feedback loops

The following system represents a simple model of a biological feedback mechanism (compare [12, 131):

x; = 1(x, f 0 - oll(O * Xl

I Xi = Xi-l - O!i(t) ‘Xi for 2 5 i 5 n

where f and CY~ are r-periodic in t 2 0 for 1 I i I n and f([, t) 2 0, (af/ay)(c, t) > 0 for c > 0, c z 0. The system is weakly positive and cooperative. The condition (af/ay)(c, t) > 0 ensures that the right-hand side is irreducible. Assume (J2f/6’[2)(c, t) < 0, for [ > 0, t 2 0, then from theorem 4 we get a limit set trichotomy for solutions of the above system. The case f(L 0 = \14* %(0, 01~ r-periodic was not covered in the literature.

Gonorrhea model

A well-known model of gonorrhea transmission (cf. [2, 9, 131) is

Xl = -ai ’ Xi + (Ci - Xi) ’ ~ Pji(t) . Xj for x, t 2 0, i E _n. j= 1


The population is divided in n groups each of constant size Ci, i E _n.

xi infectious in group i ci - Xi susceptibles in group i

Pji Ct) rate of infection (group j to group i), /9ji(t) is r-periodic (Y 2 0), nonnegative, bounded from below, and [Pji(t)] is irreducible

(yi Ct) curing rate, cui(t) is r-periodic, positive, bounded from below.

The natural domain of definition is E = (x E R”, 1 xi 5 Ci, 1 5 i 5 n). E is positively invariant for the above system.

Denote by F(x, t) the right-hand side of the gonorrhea model with components 4:(x, t) i E _n. Then F is weakly positive in E and

D,F(x, t) =

-al(t) - i Pjl(t) * Xj 0 . . . 0 j=l

0 . . . . . . . .

0

0 . . . 0 -an(t) - i Pjn(t) * Xj j=l

(Cl - Xl> * Pl,W (Cl - Xl) * P2lW f *. (Cl - x,) . P,,(t)

(c* - x2) * Pl2W + 1 L (c, - &I) * Pl,W . f *

We see that F is cooperative for x E E and since [Pji(t)] is irreducible the same holds for D,F(x, t) for x E int E and t 2 0. F is weakly concave in E since all second partial derivatives are nonpositive and at least one is negative.

Hence we can apply theorem 4 again.

A model of symbiotic interactions

A symbiotic (= mutualistic) interaction between two populations is defined as an interaction that results in a benefit for one or both of the populations. A model for symbiosis between n species is

Xii = Xi .fi(X, t) for x, t 2 0, 1 I i I n

where fi(x, -) is r-periodic and Xi is the size of the ith population i E n.

F(x, t) := (xl * fi(x, t), x2 *_A@, th . . . , xr, * fn(x, 0)

and assume that F is ray concave.

Let


Then

D,F(x, t) =

r

J-,(x, 0 + Xl * g (x, t) 1

x1 * 2 (x, t) 2

x1 * g (x, t) n

x2 - 2(x, t)

1

x, * 2 (x, t) .I& t) + xn 1

- 2 (x, t) n

and F satisfies (ci) if we assume that (afi/axj)(x, t) L 0 for i # j, and the matrix

[ I g.tx, 0 J i,j

is irreducible for x > 0, t 2 0. The assumptions of theorem 3 are satisfied. In the symbiotic model we cannot apply theorem 4 since the closure of each lower dimen-

sional part (i.e. P is a part with index set Is g) is invariant. Notice that the Lotka-Volterra model for symbiosis is not included.

1.

2. 3. 4. 5.

6. 7.

8.

9.

10.

11.

12.

13.

14.

REFERENCES

ARONSSON G. & KELLOGG R. B., On a differential equation arising from compartmental analysis, Math. Biosci. 38, 113-122 (1978). ARONSSON G. & MELLANDER I., A deterministic model in biomathematics, Math. Biosci. 49, 207-222 (1980). BAUER H. & BEAR H. S., The partmetric in convex sets, Pucif. J. math. 30, 15-33 (1969). COPPEL W. A., Stability and Asymptotic Behavior of Differential Equations. HMM, Boston (1965). HIRSCH M. W., Systems of differential equations that are competitive or cooperative II: Convergence almost everywhere, SIAM J. Math. Analysis 16, 423-439 (1985). KAMKE E., Zur Theorie der Systeme gewahnlicher Differentialgleichungen II., Acta math. 58, 57-85 (1932). KRASNOSEL’SKII M. A., Trunslution Along Trajectories of Differential Equations. American Mathematical Society, Providence, Rhode Island (1968). KRAUSE U., A nonlinear extension of the Birkhoff-Jentzsch theorem, J. math. Analysis Applic. 114, 552-568 (1986). LAJMANOVICH A. & YORKE J. A., A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci. 28, 221-236 (1976). LAKSHMIKANTHAM L. V. & LEELA S., DifSerential and Integral Inequalities, Vol. 1. Academic Press, New York (1969). MILLER M.. ijber das Fundamentaltheorem in der Theorie der gewbhnlichen Differentialgleichungen, Math. Z. 26, 619-645.(1927). SELGRADE J. F., Asymptotic behavior of solutions to single loop positive feedback systems, J. difJ Eqns 38,80-103 (1980). SMITH H. L., Cooperative systems of differential equations with concave nonlinearities, Nonfinear Analysis 10, 1037-1052 (1986). THOMPSON A. C., On certain contraction mappings in a partially ordered vector space, broc. Am. math. Sot. 14, 438-443 (1963).

APPENDIX

As in Section 3 we consider the system

x’ = F(x, t), XE IR: (*)

with F = (F F I, 2,“., F,) r-periodic in t E IR, for some 5 > 0, continuous in (x, t) E I?: x R, and continuously

differentiable in x E int R1.

PROPOSITION. Assume that F satisfies (w), (ci) and (rc). Let p(t,x) be a solution of (*) with initial condition

~(0, x) = x 2 0. Then v(t, x) is defined for all t t 0.


Proof. Let x > 0 and t 2 0. By the mean value theorem and since F is ray concave we have that

F(x, t) : D,F(x, f) . x.

Define h(s) := C:=, xie(s. x, t) for s > 0. Then with the above inequality it follows that

h’(s) := f. i x, D,fi(s. x, f) . s. x I f . i: x, ‘y(s. x, f) = f. h(s) ,=I I=,

henceh(s)isasolutionof~‘~(I/s)~z,s~Oandh(s)~h(l)~s~ /h(l)/.sforsz I.LerJJxll 2 landwritex=s~x,

with Ilx,il = 1, I/x1( = s 2 1. Let (., .) be the standard scalar product in IR”.

Using the above inequality we get that

(x, m, 0) = i: s xgj. F;(s .x0;. t) = s . h(s) 5 sz lh(l)l = 1(x,, F(x,, l))l . (x,x)

5 max(l(y, F(y, r))( ( y 2 0, l/y/( = I,0 5 f 5 rl. (x,x) =: k. (x,x).

For Ijx(I 5 1 we have that (x, F(x, t)) 2 k by definition of k and it follows that

(x, F(x, f)) 5 k + k . (x,x) for x E int a:.

Let 9(l, x) be a solution of (*) with initial condition 9(0, x) = x E int IQ: We have that

5 IIrp(f, x)11’ = 2 (cp(r, x), F(df, xl, 0) 5 2k . (yl(f, x), 9(r, x)) + 2k

with k = max(I(y, F(y, t))l ) y z 0, IJyJJ = 1, 0 5 t 5 r) and it follows that

Il9(t, x)lV 5 eZk’(llxllZ + 1) - 1

thus a solution of (*) cannot become infinite in finite time and hence 9(t, x) is defined for f 2 0.

Let y E I?: then we can find x E int IR: withy 5 x and since F satisfies (w) and (ci) we get that Ilcp(t, y)ii 5 jlcp(t, x)11

and as above 9(r,y) is defined for all t 2 0. n

Remark. The idea of the above proposition goes back to [7].


2

1.5

‘;: 1

0.5

0

r- 2

1

1.5 -

x l-

0.5 -

0 1

0 > 0 ---a. -0 e-0 o_..(__ * . . . 1

I

0 0.5 1 1.5 2

Xl

Fig. A2. Iteration plot of TX = (G, d&).

0 0.5 1 1.5 2

x

Fig. A3. Phase portrait of example 1.


x

2-

1.5 -

I-

0.5 -

O- -_ ~

0 0.5 1 1.5 2

x

Fig. A4. Phase portrait of example 2.

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A limit set trichotomy for monotone nonlinear dynamical systems