(almost periodic) cooperative systems, the property (P2) that every bounded

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1 SIAM J. MATH. ANAL. Vol. 24, No. 5, pp , September 1993 ()1993 Society for Industrial and Applied Mathematics 012 STRICTLY NONAUTONOMOUS COOPERATIVE SYSTEM WITH A FIRST INTEGRAL* BAORONG TANG?, YANG KUANG?, AND HAL SMITH? Abstract. The authors consider the nonautonomous cooperative system dxi/- Fi(t, Xl,...,xn) (i 1,...,n) in the nonnegative orthant in the real n-dimensional Euclidean space, which has the first integral with positive gradient. The authors guess that every solution to such a system either converges to an asymptotic state (for the almost periodic (or periodic) case, this state is an almost periodic (or periodic) solution) or eventually leaves any compact set. They partly prove this conjecture. Key words, cooperative systems, nonautonomous systems, almost periodic (or periodic) solution, first integral, Lyapunov function, uniformly stable, skew-product flow, w-limit.set AMS subject classifications, primary 34C27; secondary 34C25, 34D20, 90A16 1..Introduction. The autonomous gross-substitute system forms a mathematical model for the classical law of supply and demand in economics, which has been studied by many economists [9] and has the property (P1)" every bounded solution converges to an equilibrium. Such a system is a special class of cooperative systems. Cooperative (or competitive) systems are a class of dynamic systems 5c F(x), x E U C Rn, which satisfy Conditions OFi/Oxj _> 0 (_< 0) for i : j. Recently, systems of this type received much study (see references in [3]-[5] and [15]), since they have wide applications in biology and chemistry (for example, see [6], [17]). In [3]-[5] Hirsch proved many important results about such systems, one Of.which is that,.for systems that are cooperative and irreducible, almost all (with respect to the Lebesgue measure) points whose forward orbits are bounded approach the equilibrium set. But in general, the property (P1) that every bounded solution converges to an equilibrium or to a closed orbit is not true. A construction of Smale [13] shows that any dynamics is possible for such systems. Hence, one tries to find some special classes of systems of that type for which the property (P1) is true. In [14] Smillie established a particular class of cooperative systems for which the property (P) is true. In [7] Mierczyfiski found another class of cooperative systems, namely, strictly cooperative systems with a first integral, having the property (P) which generalized the result in [9]. Other relevant results can be found in Arino [19] and the references cited therein. As we know, nonautonomous systems are more realistic and more general in mathematical modelling. For example, if one considers the seasonal effects in economics, it is important to study time-dependent or nonautonomous gross-substitute systems, which are a special class of nonautonomous cooperative systems. For the case of periodic (almost periodic) cooperative systems, the property (P2) that every bounded *Received by the editors September 3, 1991; accepted for publication (in revised form) December 3, 1992.?Department of Mathematics, Arizona State University, Tempe, Arizona The research of this author was partially supported by National Science Foundation grant DMS The research of this author was partially supported by National Science Foundation grant DMS

2 1332 BAORONG TANG YANG KUANG AND HAL SMITH solution converges to a periodic (almost periodic) solution is important. Smith [16] generalized the result in [14] to the periodic case: the property (P2) holds. Nakajima [8] and Sell and Nakajima [12] studied the nonautonomous gross-substitute systems for which the property (P2) is true. In this paper we study the periodic (almost periodic) cooperative systems with a general class of first integrals. For such systems, whether the property (P2) is true remains unknown. We conjecture here that such a property continued to hold. We partially answer the above conjecture in this work. Under some conditions, we are able to show that this conjecture is indeed true. Our results generalize those of [8] and [12]. Without loss of generality, we concentrate our study on the almost periodic case. We adapt the idea similar to that in [12]" by using a Lyapunov function, we show that any "positively compact" solution Of the above system (defined in 2) is asymptotically almost periodic. The first integral used in [8] and [12] is i1 xi, for which there are many properties which played a very important role in [8] and [12]. But since we consider more general first integrals, these properties fail to be true. As we shall see, we introduce a new class of first integrals to construct proper Lyapunov functions and apply the theory of cooperative systems to obtain the desired results. These Lyapunov functions play important roles in this work. The paper is organized as follows. In the next section, we present some definitions and preliminary lemmas. In 3 we prove the main result, which partly gives an affirmative answer to the conjecture. In the last section we give an example. 2. Definitions and preliminary lemmas. Let R n denote the real n-dimensional Euclidean space with norm Ixl i--ln ixil for x (Xl,..,Xn) e R n and set R+ {x E R x _> 0}, R {x E R n "x _> 0}, 0R_ (x R "x 0 for some i} and IntR= R n \R,. +\.. + We denote x < y if xi < yi for each i, and x _<i y if x y, xi yi and xj _< yj for j i. By (.,.), we mean the usual inner product in Rn. Let H" R n + -. R be a C function. We denote grad H(p) as the gradient of H at p, which is the vector ((OU/Oxl)(p),..., (OH/Oxn)(p)). We denote Int H-(h) as the set {x Int R_ H(x) h}. C, + R n be a vector field. A first integral for F is a function H R -. R+, of class such that grad H(p) 0 at each p R_ and (grad H(p), F(t, p)) =_ O. The system of ordinary differential equations we consider takes the form Let F R R n -- (2.1) d-- F(t,x,...,xn) F(t,x), x R, 1 <_ i <_ n, where F(t,x) (F(t,x),...,Fn(t,x)) is defined and C 1 on R R. Throughout this paper, we assume that F(t, x) satisfies the following conditions (A1) If x _<i y then Fi(t,x) < F(t, y); (h2) There exists a first integral H for F such that grad H(x) > 0 for x E R and U(0) 0; (i3) F(t, x) is uniformly almost periodic in t; or (h3), F(t, x) is periodic in t with period w > 0. Remark. (hl) implies that (2.1) is cooperative. Also, the assumption that U(0) 0 is not necessary. Define the translate Fr by F(x,t) F(x, T + t), where R. By the hull, we mean the set " C1 {F T R}, where the closure is taken in the topology

3 COOPERATIVE SYSTEM WITH A FIRST INTEGRAL 1333 of uniform convergence on compact sets. It is known that is an almost periodic minimal set [2], [11], [18]. It is easy to see that every G E satisfies the conditions (A2), (A3) and that if x _<i y then G(t, x) <_ G(t, y). " Recall that a mapping r W R --. W is a flow if r is continuous, r(w, 0) w and r(r(w, s), t) r(w, s + t) for all w e W and s, t e It, where W is a topological Hausdorff space. If W is a product space W X Y, then a flow r is said to be a skew-product flow if r has the form r (o, a), or u, t) u, t), t)), where a Y T --. Y is itself a flow on Y. For each x R n and G, we denote by + o(x, G, t) the maximally defined solution of x = G(x, t) that satisfies o(x, G, 0) x. It is known that =(z, G, (v(z, a, a.) describes a (local) skew-product flow on R x " [10]. A solution o(x, F, t) of (2.1) is said to be uniformly stable if it is defined for all t _> 0 and for every e > 0 there is a i =/i(e) > 0 such that Io(x, F, T + t) o(y, F, T + t)l <_ e for all t _> 0 whenever T _> 0 and [o(x, F, T)- 0(y, F, T)[ _< 5 [101. Let V(x, y) R x Rn + (i) V(x, y) > 0 for all x, y R n with x 7 y; (ii) V(x, y) 0 if and only if x y; (iii) limlx_ul_.+oo V(x, y) +o. Then we say that V is positively definite. -. tt satisfy the following conditions. We shall use the following lemma, which is easily verified. LEMMA 2.1. Let o(&, F, t) be a positively compact solution, i.e., o(&, F, t) remains in a compact subset of R n.for all t >_ O. Assume that there is a positively definite function V(x,y) on I:t x I? e such that for all x,y l and G, one has D+V(o(x, G,t), o(y, G,t)) g O, where D+ denotes the right-hand derivative. Then o(, F, t) is uniformly stable. Theorems 2 and 5 in [10] yields the following theorem. THEOREM 2.1. Let r be the skew-product flow (2.2) on R x generated by system (2.1). Let o(]c, F, t) be a positively compact uniformly stable solution of (2.1) and let f denote the w-limit set of the motion r(&, F, t). Then f is a nonempty compact connected distal minimal set. Furthermore, if for some G J the section a(a) e a) e a} has only finitely many points, then f is an almost periodic minimal set, (x, G) e 12 the solution o(x, G, t) is almost periodic in t. The definition of a distal set can be found in [10]. and for each Recall that a compact invariant set M is minimal if and only if every trajectory There is dense in M. If 12 is minimal, then for x, y e f(f), (x, F) e f, (y, F) e 12. is a sequence tn --* +o such that o(x,f, tn) --, y. Ft. F. Since {o(y,f, tn)} is compact, without loss of generality, suppose that o(y, F, tn) z. Now Ft -. F and

4 1334 BAORONG TANG, YANG KUANG, AND HAL SMITH (z, F) G. Hence, if f is minimal, then for x, y ft(f), there is a sequence tn --* +c such that (x, F, t) --. y and (y, F, tn) --. z, where z e t(f). Let x(t) o(x, G, t), y(t) (y, G, t) be two solutions of the system x G(t, x). Suppose that they are defined on a common interval I. For t I, we define the following two subsets of (i" 1 < i < P {i x(t) >_ y(t)}, Q (i x(t) < y(t)}. 3. Partial answer. In this section we study system (2.1) which has H(x) as a first integral. In the rest of this paper, we make the following assumption: OH (A4) > 0 for i Cj. OxiOxj Define the function V(x, y) R R_ --. R+ as follows: n Y(x,y) E Ig(x" ",xi_,xi, y,+,...,y) H(xl,...,xi-,yi, y,+l,... Let V+(x,y) E[H(x,...,xi_,xi,yi+,...,Yn) H(x,...,x,-,yi,yi+,...,y)], P, V-(x,y) [H(x,,xi_,yi,yi+,...,Yn) H(x,...,Xi--l,Xi, Yi+l,...,Yn)], then V(x, y) V+ (x, y) / V-(x, y). LEMMA 3.1. Let x(t) (x, F, t), y(t) (y, F, t) be two solutions of the system (2.1) with a common interval I, then for t, t2 I with. t < t2, we have < Proof. We first prove that (3.1) V+(x(t2),y(t2)) <_ V+(x(t),y(tl)). Let zi(tl) max{xi(tl),yi(tl)}, z(tl) (zl(tl),...,zn(tl)), then z(tl) > y(tl) and z(t) > X(tl). Let z(t) (z(t),f,t- t) be a solution of (2.1), then (A1) and Kamke s theorem. (e.g., see [2]-[4] and [15]) yield that z(t) >_ y(t) and z(t) >_ x(t) fort_>t, i.e., z(t) >_ max{x(t),y(t)} (max{xl(t),y(t)},...,max{xn(t),yn(t)}) for t _> t. Thus, V+(z(t),y(t)) V(z(t),y(t)) H(z(t),...,zn(t))- H(yl(t),...,yn(t)) for t _> t. Since H is the first integral for F, we have (3.2) V(z(t),y(t)) V(z(tl),y(t)) V+(x(t),y(t)) for t _> t. Since z(t2) _> max{x(t2),y(t2)}, the assumption (A4) implies that (3.3) V(z(t2), y(t2)) _> V(max{x(t2), y(t2)}, y(t2)) V+(x(t2), y(t2)). By (3.2)and (3.3), (3.1) follows. Similar arguments show that Y-(x(t2),y(t2)) <_ Y-(x(t),y(t.)) and the proof is thus completed. Remark. Clearly, if V(x, y) defined above satisfies that limlz_ul_+o V(x, y) +cx3i then V is positively definite.

5 COOPERATIVE SYSTEM WITH A FIRST INTEGRAL 1335 LEMMA 3.2. In (2.1), in addition to (A1), (A2), (A3) (or (h3) ), and (A4), assume that limlx_ul_,+o V(x, y) +oc and.for every h 6 R+, there is at most one bounded solution which is defined and belongs to H-1 (h).for all t R. Then.for every positively compact solution &(t), solution) such that (3.4) lim I&(t) (t)l 0. there exists an almost periodic solution (or periodic Proof. Let &(t) o(&, F, t) be a positively compact solution and limit set of the corresponding motion r(&, F, t) in R_ x. Let t(f) (x e R (x, F) gl}, then Lemma 3.1, combining Lemma 2.1 and Theorem 2.1, implies that t(f) consists of a single point y and the solution (t) o(y, F, t) is almost periodic. Now we prove (3.4). By Lemma 3.1, one - has D+Y(&(t),(t)) <_ 0 for all t >_ 0. Thus, the limit limt-,+o V(&(t), (t)) exists. Choose a sequence tn (y,f) and (2c, F, tn) z. Since Ftn --. F it follows that z 12(F) and so z y. Thus lim_+ V(&(t), (t)) 0 and so lim,+o For y (yl,... Yn) Rn, we define Sk(y) {x (x,...,xn) e R xi <_ yi, i= 1,...,k- 1, Xk < Yk and xk+ > Yk+, Xi >_ yi for i k + 2,...,n}, k-- 1,...,n- 1. n--1 LEMMA 3.3. Let x(t), y(t) be two solutions of (2.1). If x(t) tjk= Sk(y(t)), then dv(x(t),y(t)) Proof. We consider the case x(t) e S(y(t)); for the other cases, the proofs are similar. If x(to) S(y(to)), then there is a 5 > 0 such that x(t) Sl(y(t)) for to -5 < t < to + 5. Hence, for t 6 (to 5, to + 5), V(x(t), y(t)) H(yl (t), yn(t)) + H(xl (t), 2H(x (t), y2(t),..., yn(t)). Since H(xl (t), xn(t)) H(yl (t), yn(t)) constant, we have Notice that dv(x(t),y(t)) -20H(x (t), y2(t),..., y(t)) FI (t, x (t),..., x(t)) 2 E OH(x (t), y2(t),..., y(t)) Fi(t, y (t),..., y(t)). =2 OH(x (t), y2(t),..., yn(t)) F(t,x(t), y2(t),..., yn(t)) OXl + OH(l(t),(t),...,,(t))Fi(t,x(t),(t),...,,(t)) 0 i=2 Oy

6 1336 BAORONG TANG YANG KUANG AND HAL SMITH and so dv(x(t),y(t)) Now assumption (A1) yields that -20H(xl(t), y2(t),..., yn(t)) x (F(t,x(t),x2(t),...,x(t))- F(t,x(t),y2(t),...,y(t))) 2 E OH(x (t), y2(t),..., yn(t)) Oyi i=2 (F(t, yl(t),y2(t),...,y(t))- F(t,x(t),y2(t),...,y(t))). dy(x(t),y(t)) <0. E3 For any two points x, y R=, which are not related, i.e., that x < y or y < x is not true, we always find a map P R R= such that Px S(y) for some k e {1,...,n- 1}. Let T be the set of such maps. Since R n is finite dimensional, T is finite. Define Vp(x, y) V(Px, Py) for P e T, and let W(x, y)- Y]PeT Vp(x, y), then we have LEMMA 3.4. Let x(t), y(t) be two solutions of (2.1) with H(x(t)) H(y(t)). Then (3.5) Proof. dw(x(t),y(t)) Let P E T. Consider the system du Fp(t, u), where u Px, Fp(t, u) P(F(t, p-lu)). It is easy to check that system (3.5) satisfies (A1) to (A4). By Lemma 3.1 and Lemma 3.2, we have D+Y(Px(t),Py(t)) <_ 0 n--1 and if Px(t) k=l Sk(Py(t)) then dv(p(t),pu(t)) < 0. Hence D+Vp(x(t),y(t)) < 0 and n-i dvp(x(t), y(t)) < 0 for Px(t) U Sk(Py(t)). For x(t), y(t) with H(x(t)) H(y(t)), (A2) implies that x(t) and y(t) are not related and so we can choose P0 T such that Pox(t) Sk(Poy(t)) for some k, then the above argument shows that dvpo (X(t), y(t) <0. Now dw(x(t), y(t)) dvp(x(t), y(t)) E <0. El PT We are now in a position to prove our main result. =1

7 COOPERATIVE SYSTEM WITH A FIRST INTEGRAL 1337 THEOREM 3.1. In (2.1), in addition to (A1), (A2), (A3) (or (A3) ), and (A4), assume that limlx_ul_,o V(x, y) +x). Then.for every positively compact solution &(t), there exists an almost periodic solution (or periodic solution) such that lim ]&(t) (t)[ 0. Proof. Let &(t) qo(&, F, t) be a positively compact solution and f the w-limit set of the corresponding motion 7r(&, F, t) in R. Let 12(F) {x e R_: (x, F) e f}. Suppose f(f) contains more than one point. Let x0, y0 E f(f) with xo y0 and x(t) qo(xo, F, t), y(t) (yo, F, t) be the corresponding solution of (2.1). Since x(t) and y(t) stay in a compact set in R_, they are defined for all t E R. By Theorem 2.1, f is -a distal set. Ellis s theorem [1] implies that the product flow on 2 f is the union of minimal sets. Hence there is a sequence tn --* +oc such that x(tn) --* xo and y(tn) yo. Lemma 3.4 implies that there is a T R (say T > 0) such that u(t)) _< < W( o, no) t _> Letting tn -- -t-cx3, one gets the contradiction W(xo, yo) lira W(x(tn), y(tn)) < W(xo, Yo). For T _< 0, a suitable translation of x(t) and y(t) also yields a contradiction. Thus, the proof follows from Lemma 3.2. [3 Remark. If we take H(xl,...,xn) xl / / xn, it is trivial that (A4) is satisfied. Hence the above result generalizes those of [8] and -- [12]. In the rest of this section, we suppose that (2.1) is periodic, i.e., (A3) is true and (Ai), (A2) and (A4) hold. Define the Poincard map T R_ Tx (x, F, w). R as follows Forx0 Rn+, definex0/r_ {x0+x" x R}. Then it is easy to prove the._ following lemma. LEMMA 3 5 Let xo Int R n be a periodic point, i.e., the solution qo(xo, F, t) is periodic with the period w. Then xo + R n + is positively invariant under T, i.e. for x E (xo + R)- xo, Tx Int (x0 + R). Thus, xo is a unique periodic point on xo + OR?. LEMMA 3.6. Let xo IntR be a periodic point. Then, for every e > 0 there exists > 0 such that for each h e [H(c) 5, H(c) + 5] fq [0, M), where M is the least upper bound of the values of H, there is a periodic point Xh such that H(xh) h, Xh > xo for h > H(xo) (respectively, Xh < Xo for h < H(xo)) and IXh xol < e. Proof. We consider the case h > H(xo). Let ei (0,... 0, il, 0,... 0), then from (A2), we have H(xo + ei) > H(xo). Let (1 min<i<n{h(xo + ei)- H(x0)}, then (A2) implies that U-X(h)fq (xo + pei) for all 1 _< i < u, p _> 0, and h IS(x0), U(x0)+(ti/2)]. It is easy to see that U-X(h)(zo+rt?) is homeomorphic to the (n- 1)-dimensional disk. Lemma 3.2 and the definition of H shows that T maps H-(h) N (xo + R_) to itself. The Brouwer fixed-point theorem yields that there is a periodic point Xh U-(h)fq (xo + R_) for all h [U(xo),U(zo)+ (t/2)]. Also Xh XO.

8 1338 BAORONG TANG, YANG KUANG, AND HAL SMITH Clearly, we can choose a point x > x0 such that for all x E H-(H(xo)+ (5/2))Cl(x0d-R), x < x. Then for all x U-(h) f(xod-r) with h [H(xo),U(xo)- (5/2)], we have x0 _< x < xl. Let m min(ou(x)/oxi" 1 <_ <_ n, x0 _< x _< x }. If we choose 5 < min(me, (5/2)}, then it is easy to conclude that for h IS(x0), U(xo) d- ) [0, M), we have Ix- x01 < e for x H-(h), in particular, IXh- XOI ( e. THEOREM 3.2. In (2.1) assume that the assumptions made in Theorem 3.1 are true. Then the set of periodic points in Int R n / is connected. Proof. It follows from Lemma 3.6 and Theorem An example. Consider the following three-dimensional system in 1 3" +. (4.1) &2 c(t)xl x2 + x3 2a(t)x2 + b(t)x3 3+x +x3 + 2/xl -t-x2 where a(t) > O, b(t) > 0,. c(t) > O, a(t), b(t) and c(t) are almost periodic. Clearly, U(xl,x2,x3) (Ux)(2 + x2) + (1 + x)(1 + x3) + (1 + x2)(2 + x3) 5 is the first integral and system (4.1) satisfies (A1), (A2), (A3), and (A4). Also, is positively definite. Hence, we can apply Theorem 3.1 to system (4.1). REFERENCES [1] R. ELLIS, Distal transformation groups, Pacific J. Math., 8 (1958), pp [2] A. M. FINK, Almost periodic differential equations, Lecture Notes in Math. 377 (1974), Springer- Verlag, New York. [3] M. W. HmSCH, Systems of differential equations which are competitive or cooperative I: Limit sets, SIAM J. Math. Anal., 13 (1982), pp [4] Systems of differential equations that are competitive or cooperative II: Convergence almost everywhere, SIAM J. Math. Anal., 16 (1985), pp [5] The dynamical systems approach to differential equations, Bull. Amer. Math. Soc. (N.S), 11 (1984), pp [6] D. S. LEWNE, Qualitative theory of a third order nonlinear system with examples in population dynamics and chemical kinetics, Math. Biosci., 77 (1985), pp [7] J. MIErtCZYISKI, Strictly cooperative systems with a first integral, SIAM J. Math. Anal., 18 (9s7), pp [8] F. NAKAJIMA, Periodic time-dependent gross-substitute systems, SIAM J. Appl. Math., 36 (1979), pp [9] F. NIKAIDO, Convex Structure and Economic Theory, Academic Press, New York, [10] R. J. SACKEa AND G. R. SELL, Lifting properties in skew-product flows with applications to differential equations,, Mere. Amer. Math. Soc. 190, [11] G. R. SELL, Nonautonomous differential equations and topological dynamics, Trans. Amer. Math. Soc., 127 (1967), pp

9 COOPERATIVE SYSTEM WITH A FIRST INTEGRAL 1339 [12] G. R. SELL AND F. NAKAJIMA, Almost periodic gross-substitute dynamical systems, T6hoku Math. J., 32 (1980), pp [13] S. SMALE, On the dierential equations o] species in competition, J. Math. Biol., 3 (1976), pp [14] J. SMILLIE, Competitive and cooperative tridiagonal systems of differential equations, SIAM J. Math. Anal., 15 (1984), pp [15] H. L. SMITH, Systems of ordinary differential equations which generate an order preserving flow. A survey of results, SIAM Rev., 30 (1988), pp [16] Periodic tridiagonal competitive and cooperative systems of differential equations, SIAM J. Math. Anal., 22 (1991), pp [17] B. TANG, On the existence of periodic solutions in the limited explodator model ]or the Belonsov-Zhabotinskii reaction,, Nonlinear Anal., TMA, 13 (1989), pp [18] T. YOSHIZAWA, Stability theory and the existence o] periodic and almost periodic solutions, Lectures in Appl. Math., 14, Springer-Verlag, New York, [19] O. ARINO, Monotone semi-flows which have a monotone first integral, preprint.