Dynamics of nonautonomous tridiagonal. competitive-cooperative systems of differential equations

Transcription

1 Dynamics of nonautonomous tridiagonal competitive-cooperative systems of differential equations Yi Wang 1 Department of Mathematics University of Science and Technology of China Hefei, Anhui, , P. R. China July 13, supported by FANEDD of China and NSF of China.

2 Abstract. Skew-product flow which is generated by a nonautonomous recurrent tridiagonal competitive-cooperative system of differential equations is considered. It is shown that any minimal set is an almost 1-1 extension of the base flow and any ω-limit set contains at most two minimal sets, which generalizes the results of J. Smillie [SIAM J. Math. Anal., 15(1984), pp ] in autonomous cases and H. L. Smith [SIAM J. Math. Anal., 22(1991), pp ] in timeperiodic cases. Further results are also obtained in the case that the base flow is almost automorphic or almost periodic. Key words: Nonautonomous tridiagonal systems, Competitive-cooperative; Skew-product flows Mathematics subject classification: 34C12, 34C27, 37B55, 37N25, 92D25 1 Introduction The current paper is devoted to the study of the nonautonomous tridiagonal system ẋ 1 = f 1 (t, x 1, x 2 ), ẋ i = f i (t, x i 1, x i, x i+1 ), 2 i n 1; (1.1) ẋ n = f n (t, x n 1, x n ), where f = (f 1, f 2,, f n ) is defined on R R n and satisfies the following condition: (F1) f is C 1 -admissible; i.e., f(t, x) = (f 1 (t, x 1, x 2 ),, f i (t, x i 1, x i, x i+1 ),, f n (t, x n 1, x n )), together with its first derivatives with respect to x = (x 1, x 2,, x n ), is bounded and uniformly continuous on R K for any compact set K R n. We also assume that there are ε 0 > 0 and δ i { 1, +1}, such that δ i f i f i+1 (t, x) ε 0, δ i (t, x) ε 0, 1 i n 1, (1.2) x i+1 x i for all (t, x) R R n. This assumption implies that the Jacobian matrix f/ x, corresponding to (1.1), is tridiagonal and sign symmetric in the sense that f i / x i+1 and f i+1 / x i have the same 1

3 sign δ i. If δ i = 1 for all i, then (1.1) is called competitive. If δ i = 1 for all i, then (1.1) is called cooperative. We introduce new variables, following Smith [23]. We let ˆx i = µ i x i, µ i {+1, 1}, 1 i n, with µ 1 = 1, µ i = δ i 1 µ i 1. Then the system (1.1) transforms into a new system of the same type with new ˆδ i = µ i µ i+1 δ i = µ 2 i δi 2 = 1. Therefore we can always assume, without loss of generality, that the competitive-cooperative system (1.1) is in fact cooperative and (F2) f i x i+1 (t, x) ε 0, f i+1 x i (t, x) ε 0, 1 i n 1, (t, x) R R n. The system of equations (1.1) with assumption (F1)-(F2) generates a monotone dynamical system. There is an extensive literature on monotone dynamical systems, starting with the work of Hirsch [8] for monotone semiflows. The results of Hirsch and later improvements by Matano [10], Smith and Thieme [24, 25], and Poláčik [13] established that most orbits of a strongly monotone semiflow converge to equilibria. For strongly monotone smooth mapings, Poláčik and Tereščák [14], and Tereščák [26] proved that the forward orbits are generically convergent to cycles. Shen and Yi [17] further developed the theory to strongly monotone skew-product semiflows and proved the almost automorphy of the linearly stable minimal sets of such skew-product semiflows with an almost periodic minimal base flow. However, the generic convergence property for continuous- and discrete-time in strongly monotone dynamical systems failed in almost periodic systems even within the category of almost automorphy (see [17]). For system (1.1), if f is independent of t, the well-known result by Smillie [22] showed that all bounded trajectories converge to equilibria. Smith [23] studied the time-periodic system (1.1) and proved that every bounded solution is asymptotic to a T -periodic solution if f is time periodic with period T > 0. Zhao [28] has generalized their results to asymptotic periodic differential equations. They both used an integer-valued Lyapunov function to prove their results. Later on, Fiedler and Gedeon [2] introduced another real valued Lyapunov function in autonomous systems and prove the same convergent results and applied them to tridiagonal cooperative-competitive Kolmogorov 2

4 systems. See also Freedman and Smith [4], Gyllenberg and Wang [5] for extensions of this work. In nature, populations evolve influenced by external effects which are roughly, but not exactly periodic, or under environmental forcing which exhibits different, noncommensurate periods. This sort of time dependence can arise from the interplay of short-term weather cycles and seasonal climate variations, or from the superposition of daily and annually periodic phenomena, and so on. Models with such time dependence are characterized more appropriately by quasi-periodic or almost periodic equations or even by certain nonautonomous equations rather than by periodic ones. Time nonperiodic equations are therefore worth studying. The current paper is devoted to study the dynamics of the nonautonomous equations (1.1)- (1.2). We shall employ the notions of skew-product semiflows and the abstract theory of monotone dynamical systems to carry out our study. To be more specific, consider (1.1) and embed it into the skew-product flow Π t : R n H(f) R n H(f), Π t (x, g) (π(t; x, g), g t), (1.3) where π(t; x, g) is the solution of ẋ 1 = g 1 (t, x 1, x 2 ), ẋ i = g i (t, x i 1, x i, x i+1 ), 2 i n 1; (1.4) ẋ n = g n (t, x n 1, x n ), with π(0; x, g) = x, x R n, and g = (g 1,, g n ) H(f), (g t)(, ) = g(t +, ). H(f) := cl{f τ τ R, f τ(t, u) = f(t + τ, u)}, where the closure is taken in the compact open topology (see section 2 for more detail on H(f)). Obviously, π satisfies the cocycle property, i.e, π(t + s; x, g) = π(s; π(t; x, g), g t) for all s, t R and g H(f). Furthermore, it is also easy to check that (F1) and (F2) holds for any g H(f), namely, (G1) g is C 1 -admissible; (G2) g i x i+1 (t, x) ε 0, g i+1 x i (t, x) ε 0, 1 i n 1, (t, x) R R n, 3

5 are satisfied for all g H(f). Throughout this paper we will always assume that H(f) is recurrent or minimal. This is satisfied, for instance, when f is a uniformly almost periodic, or, more generally, a uniformly almost automorphic function; i.e., when it is admissible and almost periodic or almost automorphic (see section 2 for more detail). In the terminology of the skew-product flow (1.3), the study of asymptotic behavior for a bounded solution π(t; x, f) of (1.1) with (F1)-(F2) then gives rise to the problem of understanding the ω-limit set ω(x, f) of the bounded orbit Π t (x, f) in R n H(f). In particular, in the case that f is time periodic with period T > 0, it is well known that each ω-limit set ω(x, g)(g H(f) S 1 ) is a periodic minimal set in R n H(f) with period T (see [23]) (in the autonomous case, each ω-limit set is an euqilibrium, see [22]). Nevertheless, similar results are false in general for time nonperiodic equations (1.1)-(1.2), namely, one can not always expect an ω-limit set ω(x, g) to be a 1-cover (see definition in section 2) of the base flow on H(f). There are examples even in almost periodic scalar ODEs (n = 1 in (1.1)) which suggest that the ω-limit set ω(x, g) may not be minimal (see [15]), the ω-limit sets ω(x, g) may contain two minimal sets (see [9]), and the ω-limit set ω(x, g) may not be 1-cover of the base even if ω(x, g) is minimal (see [3, 9]). Our focus in this paper is on the structure of the ω-limit sets and the minimal sets for the skew-product flow (1.3) generated by (1.1)-(1.2). We shall prove that any minimal invariant set of (1.3) is an almost automorphic extension of H(f) (i.e., an almost 1-cover of H(f), see definition in section 2), and every ω-limit set ω(x, g) of (1.3) contains at most two minimal sets. Therefore, for time almost automorphic (periodic) equations (1.1)-(1.2), the existence of the almost automorphic solutions is obtained and the frequency module of any almost automorphic solution is contained in that of f. We also discuss cases in which (1.3) admits almost periodic minimal ω-limit sets. Our results here are natural generalization of the results of Smillie [22] and Smith [23]. Moreover, in a certain sense, our results also generalize the results in spatially homogeneous cases by Hetzer and Shen [6], who investigated the dynamics of two-dimensional competitive or cooperative almost periodic systems. See also [7] and [21] for extensions of this work. 4

6 As in [22, 23, 28], the integer-valued Lyapunov function developed in [22] play important roles in our current investigation. The idea of using integer-valued Lyapunov function seems to go back to the work of Nickel [12] and later to that of Matano [11] (called lap-number) and Angenent [1] (called zero number). The zero number has been used by Shen and Yi ([17]-[20]) to establish the dynamics for almost periodic scalar parabolic equations. This paper is organized as follows. In Section 2 we agree on some notations, give relevant definitions and preliminaries which will be important to our proofs. We investigate the lifting properties of the minimal sets of (1.3) in Section 3. Section 4 is devoted to the study of the structure of ω-limit sets. In this section, we also discuss some cases in which (1.3) admits almost periodic minimal ω-limit sets. 2 Notations and preliminary results In this section, we summarize some preliminary materials to be used in later sections. First, we give a brief review about almost periodic and almost automorphic functions. We then summarize some lifting properties of compact dynamical systems. Finally, we give some basic properties of solutions of (1.4) for later use. Definition 2.1. (1) A function f C(R, R n ) is almost periodic if, for any ε > 0, the set T (ε) := {τ : f(t + τ) f(t) < ε, t R} is relatively dense in R. f is almost automorphic if for any {t n} R there is a subsequence {t n } and a function g : R R n such that f(t + t n ) g(t) and g(t t n ) f(t) hold pointwise. (2) A function f C(R D, R n )(D R m ) is uniformly almost periodic or uniformly almost automorphic in t if f(t, u) is bounded and uniformly continuous on R K for any compact subset K D, (i.e., f is admissible), and is almost periodic or almost automorphic in t R. (3) Let f C(R R n, R n ) be uniformly almost periodic (almost automorphic). Then H(f) = cl{f τ : τ R} is called the hull of f, where f τ(t, x) = f(t + τ, x) and the closure is 5

7 taken under the compact open topology. Moreover, H(f) is compact and metrizable under the compact open topology (see [16, 17]). (4) Let f C(R R n, R n ) be uniformly almost periodic (almost automorphic), and f(t, x) λ R a λ (x)e iλt (2.1) be a Fourier series of F (see [27, 17] for the definition and the existence of Fourier series). Then S = {λ : a λ (x) 0} is called the Fourier specturm of f associated to the Fourier series (2.1), and M(f) = the smallest additive subgroup of R containing S(f) is called the frequency module of f. Moreover, M(f) is a countable subset of R (see [17]). Lemma 2.1. Let f(t, x) C(R R m, R n ), g(t, x) C(R R l, R k ) be two uniformly almost automorphic functions. Then M(g) M(f) if and only if for any sequence {α n } R, if lim n f(t + α n, x) = f(t, x) uniformly for t and x in bounded sets, then lim n g(t + α n, x) = g(t, x) uniformly for t and x in bounded sets. Proof. See [17]. Let Y be a compact metric space with metric d Y, and σ : Y R Y, (y, t) y t be a continuous flow on Y, denoted by (Y, σ) or (Y, R). If (Z, R) is another continuous flow, a flow isomorphism from (Y, σ) to (Z, R) is a continuous bijective mapping π from Y to Z such that π(y t) = π(y) t for all y Y and t R. A subset S Y is invariant if σ t (S) = S for every t R. A subset S Y is called minimal if it is compact, invariant and the only non-empty compact invariant subset of it is itself. We say that the continuous flow (Y, σ) is recurrent or minimal if Y is minimal. For a given net α = {t n } R and y Y, define T α y = lim n y t n provided that the limit exists. Definition 2.2. (1) A point y 0 Y is called an almost periodic point if for any nets α, β in R, there are subnets α, β such that T α T β y 0 t = T α+β y 0 t uniformly in t R. 6

8 (2) A point y 0 Y is called an almost automorphic point if for any nets α in R, there is a subnet α such that T α T α y 0 t = y 0 t pointwise in t R. (3) (Y, σ) is almost periodic (almost automorphic) minimal if it is minimal and contains an almost periodic (almost automorphic) point. Remark 2.1. An almost periodic minimal flow is clearly almost automorphic minimal. If (Y, σ) is almost periodic minimal, then any y Y is an almost periodic point. If (Y, σ) is almost automorphic minimal, then the set of almost automorphic points of Y is residual. Lemma 2.2. (1) If f C(R R n, R n ) is uniformly almost periodic (almost automorphic), then the time translation flow (H(f), R), (g, t) g t for g H(f) and t R, is almost periodic (almost automorphic) minimal. (2) If f C(R R n, R n ) is uniformly almost automorphic, then for any uniformly almost automorphic function g H(f) (there are residually many by Remark 2.1), M(g) = M(f). Proof. See [16, 17]. Definition 2.3. Let (Y, R), (Z, R) be two continuous compact flow. Z is called a 1-cover (almost 1-cover, or almost automorphic extension) of Y if there is a flow homomorphism P : X Y such that P 1 (y) is a singleton for any y Y (for at least one y Y ). Definition 2.4. Let E be an invariant set of (1.3). For any g H(f), a pair (x, g), (y, g) E P 1 (g) is said to be (positive, negatively) distal if inf (t R +,t R )t R 1 π(t; x, g) π(t; y, g) > 0. The pair (x, g), (y, g) is called (positive, negatively) proximal if it is not (positive, negatively) distal. E is said to be distal if any (x, g), (y, g) E(x y) forms a distal pair. 7

9 Now we focus on the competitive-cooperative system (1.4) (g H(f)) with (G1) and (G2). Following [23], we define a unique continuous function σ : Λ {0, 1, 2,, n 1} on Λ = {v R n : v 1 0, v n 0 and if v i = 0 for some i, 2 i n 1, then v i 1 v i+1 < 0} by σ(v) = #{i : v i v i+1 < 0}. Here # denotes the cardinality of the set. Note that Λ is open and dense in R n and Λ is the maximal domain on which σ is continuous. Consider the linear system dx 1 = a 11 (t)x 1 + a 12 (t)x 2 dt dx j dt = a jj 1(t)x j 1 + a jj (t)x j + a jj+1 (t)x j+1, 2 j n 1, dx n dt = a nn 1 (t)x n 1 + a nn (t)x n, where the functions a ij ( ) are continuous and defined on a nontrivial interval J and (2.2) a jj+1 (t) > 0 on J, 1 j n 1, a jj 1 (t) > 0 on J, 2 j n. (2.3) Lemma 2.3. If x(t) is a nontrivial solution of (2.2)-(2.3) on J then (i) x(t) Λ except for isolated values of t. (ii) If x(s) / Λ for some s IntJ then σ(x(s+)) < σ(x(s )). Proof. See [23]. Lemma 2.4. For any g H(f), Let π(t; x, g) and π(t; ˆx, g) be distinct solutions of (1.4) on an interval I. Then (1) π(t; x, g) π(t; ˆx, g) Λ except for finitely many values of t I. 8

10 (2) σ(π(t; x, g) π(t; ˆx, g)) is locally constant and strictly decreases as t increases through a value s at which it is not defined. (3) If I (0, + ), then there exists some T > 0 such that σ(π(t; x, g) π(t; ˆx, g)) constant for all t T. (4) If I (, 0), then there exists some T 1 > 0 such that σ(π(t; x, g) π(t; ˆx, g)) constant for all t T 1. Proof. x(t) = π(t; x, g) π(t; ˆx, g) satisfies the linear system (2.2)-(2.3), where a ij (t) = 1 0 g i x j (t, u i 1 (s, t), u i (s, t), u i+1 (s, t))ds with u j (s, t) = s(π(t; x, g)) j + (1 s)(π(t; ˆx, g)) j, j = i 1, i, i + 1. Then the results follows from (G2), Lemma 2.3 and the definition of the function σ. 3 Lifting properties of minimal sets Recall that we always assume that H(f) is minimal (or recurrent). By Lemma 2.1, this is satisfied when f is a uniformly almost periodic or a uniformly almost automorphic function. In this section, we shall prove that any minimal invariant set of the skew-product flow (1.3) is an almost automorphic extension of H(f). Motivated by [17, 18, 19], in order to do this, we first introduce a new ordering on fibres of the minimal sets. Definition 3.1. Let K R n H(f) be an invariant compact set. Then, for each g H(f), we define a partial ordering on P 1 (g) K as follows: (x, g), (y, g) P 1 (g) K and (x, g) g (y, g) if and only if there is a T > 0 such that (π(t; x, g) π(t; y, g)) 1 0 for all t T. Here ( ) 1 means the first coordinate in R n. 9

11 As usual, we say Then we have the following (x, g) > g (y, g) (x, g) g (y, g) and x y. Lemma 3.1. Let K R n H(f) be an invariant compact set. Then, for each g H(f) and (x, g), (y, g) P 1 (g) K, (1) (x, g) > g (y, g) if and only if there exists a T > 0 such that (π(t; x, g) π(t; y, g)) 1 > 0 for all t T. (2) g is a total ordering on P 1 (g) K. Proof. First observe that t π(t; x, g) can be defined on R for any (x, g) K, since K is compact and invariant. If (x, g) > g (y, g) then, by definition 3.1, (π(t; x, g) π(t; y, g)) 1 0 for all t sufficiently large. Suppose that there exists a sequence {t n }, t n, such that (π(t n ; x, g) π(t n ; y, g)) 1 = 0 for all n. Then π(t n ; x, g) π(t n ; y, g) / Λ for all n N, which contracts Lemma 2.4. The sufficiency of (1) is obvious from the uniqueness of the solutions. For any (x, g), (y, g) P 1 (g) K, Lemma 2.4(3) implies that (π(t; x, g) π(t; y, g)) 1 0 for all t T. Then (π(t; x, g) π(t; y, g)) 1 > 0 for all t T, or (π(t; y, g) π(t; x, g)) 1 > 0 for all t T. Hence, by Lemma 3.1(1), one has (x, g) > g (y, g) or (y, g) > g (x, g). Definition 3.2. Let K R n H(f) be an invariant compact set. Then, for each g H(f), a fiberwise strong ordering g on P 1 (g) K is defined as follows: (x 1, g) g (x 2, g) if and only if there are neighborhoods N 1, N 2 P 1 (g) K of (x 1, g), (x 2, g) respectively such that (x 1, g) > g (x 2, g) for all (x i, g) N i (i = 1, 2). Definition 3.3. (x 1, g), (x 2, g) K form a strongly order preserving pair if (x 1, g), (x 2, g) are fiberwise strongly ordered, say (x 1, g) g (x 2, g), and there are neighborhoods U 1, U 2 of (x 1, g), (x 2, g) in K respectively such that whenever (y 1, g), (y 2, g) K P 1 (g), (y 1, g) (y 2, g), and Π T (y i, g) U i (i = 1, 2) for some T > 0, then (y 1, g) > g (y 2, g). 10

12 The following Porposition is due to Shen and Yi [17, Theorem 2.3.1]. Proposition 3.1 (Shen and Yi). Let E R n H(f) be a minimal set of Π. Then there exists a residual and invariant set H (f) H(f) such that, for any g H (f), E P 1 (g) admits no strongly order preserving pair. Theorem 3.1. Let E R n H(f) be a minimal set of Π. Then the following holds: (a) E is an almost automorphic extension (or almost 1-cover) of H(f); (b) E is almost automorphic if and only if H(f) is almost automorphic. (c) Let f in (1.1) with (F2) be uniformly almost automorphic and let (x, g) E be an automorphic point (there are residually many), then π(t; x, g) is a (uniform) almost automorphic solution of (1.4), and moreover M(π( ; x, g)) M(f). Proof. For (a), let H (f) be the residual set in Proposition 3.1. Suppose there exists some g 0 H (f) such that card(e P 1 (g 0 )) > 1. Then we define σ(g 0 ) := min (x 1, g 0), (x 2, g 0) E P 1 (g 0) { } inf σ(π(t; t>0 x1, g 0 ) π(t; x 2, g 0 )). (3.1) (x 1, g 0) (x 2, g 0) By virtue of the definition σ and Lemma 2.4(3), there exist (ˆx 1, g 0 ), (ˆx 2, g 0 ) E P 1 (g 0 ) and some T > 0 such that σ(g 0 ) = σ(π(t; ˆx 1, g 0 ) π(t; ˆx 2, g 0 )) for all t T. (3.2) By Lemma 3.1, we can assume without loss of generality that (ˆx 1, g 0 ) > g0 (ˆx 2, g 0 ) and (π(t; ˆx 1, g 0 ) π(t; ˆx 2, g 0 )) 1 > 0 for all t T. Now we first claim that (ˆx 1, g 0 ) g0 (ˆx 2, g 0 ). Indeed, fix the T > 0 in (3.2), it then follows from the continuity of σ in Λ that there are neighborhoods Ñ1, Ñ2 of (ˆx 1, g 0 ), (ˆx 2, g 0 ) in E respectively such that (i). σ(π(t ; x 1, g) π(t ; x 2, g)) σ(g 0 ); (ii). (π(t ; x 1, g) π(t ; x 2, g)) 1 > 0 (3.3) 11

13 for all (x i, g) Ñi(i = 1, 2). Let N i = Ñi P 1 (g 0 ), (i = 1, 2). Then one has (i). σ(π(t ; x 1, g 0 ) π(t ; x 2, g 0 )) σ(g 0 ); (ii). (π(t ; x 1, g 0 ) π(t ; x 2, g 0 )) 1 > 0 (3.4) for all (x i, g 0 ) N i (i = 1, 2). Suppose that there are some (x i, g 0 ) N i (i = 1, 2) with (π(t 1 ; x 1, g 0 ) π(t 1 ; x 2, g 0 )) 1 0 for some T 1 > T. Then, from (3.4)(ii), one can find some T 2 (T, T 1 ] such that (π(t 2 ; x 1, g 0 ) π(t 2 ; x 2, g 0 )) 1 = 0. Hence, π(t 2 ; x 1, g 0 ) π(t 2 ; x 2, g 0 ) / Λ. By Lemma 2.4(2) and (3.4)(i), σ(g 0 ) = σ(π(t ; x 1, g 0 ) π(t ; x 2, g 0 )) > σ(π(t 1 ; x 1, g 0 ) π(t 1 ; x 2, g 0 )), which contracts the minimum definition of σ(g 0 ) in (3.1). As a consequence, (π(t; x 1, g 0 ) π(t; x 2, g 0 )) 1 > 0 for all t T and (x i, g 0 ) N i (i = 1, 2). This imples, by Lemma 3.1(1), that (x 1, g 0 ) > g (x 2, g 0 ) for all (x i, g 0 ) N i (i = 1, 2). Thus we have proved the claim, i.e., (ˆx 1, g 0 ) g0 (ˆx 2, g 0 ). Moreover, we will prove that (ˆx 1, g 0 ), (ˆx 2, g 0 ) forms a strongly ordered pair. To end this, we recall the neighborhoods Ñ1, Ñ2 obtained above. Let (x i, g 0 ) E P 1 (g 0 ) with Π t0 (x i, g 0 ) Ñi(i = 1, 2) for some T < t 0 < 0. Then it follows from (3.3) and the cocycle property of π that (i). σ(π(t + t 0 ; x 1, g 0 ) π(t + t 0 ; x 2, g 0 )) σ(g 0 ); (ii). (π(t + t 0 ; x 1, g 0 ) π(t + t 0 ; x 2, g 0 )) 1 > 0. (3.5) By the same arguments as above, one can obtain that (π(t; x 1, g 0 ) π(t; x 2, g 0 )) 1 > 0 for all t t 0 + T, which implies that (x 1, g 0 ) > g (x 2, g 0 ). Thus, (ˆx 1, g 0 ), (ˆx 2, g 0 ) forms a strongly ordered pair, contracting Proposition 3.1. We have proved (a). The proofs of (b) and (c) are very similar with those in [17, P.61]. For the reader s convenience we supply the proofs. For (b), the almost automorphy of E of course implies that of H(f). On the other hand, if H(f) is almost automophic, then by remark 2.1, the set Y 0 of almost automorphic points of H(f) is residual. Then Y 0 H (f) is residual and each singleton in P 1 (g) E(g Y 0 H (f) is an almost automorphic point of Π, which completes the proof of (b). For (c), if (x, g) E is an almost automorphic point, then g H(f) is an almost automorphic point in H(f) and hence P 1 (g) E = (x, g). It follows that for any net α R, T α g = g, then T α π( ; x, g) = π( ; x, g). 12

14 By Lemma 2.1 and Lemma 2.2(2), one has M(π( ; x, g)) M(g) = M(f), which completes the proof. 4 Structure of ω-limt sets In this section, we focus on the structure of the limit sets. We will prove that every ω-limit set ω(x, g) of (1.3) contains at most two minimal sets, also we will consider the case in which (1.3) admits almost periodic minimal limit sets. Again motivated by [18, 19], we proceed as the following lemma. Lemma 4.1. Fix g, g H(f). Let (x i, g) P 1 (g), (x i, g ) P 1 (g ) (i = 1, 2, x 1 x 2, x 1 x 2 ) be such that Π t (x i, g) is defined on R + (resp. R ) and Π t (x i, g ) is defined on R. If there exists a sequence {t n }, t n + (resp. t n ) as n, such that Π tn (x i, g) (x i, g ) as n (i = 1, 2), then σ(π(t; x 1, g ) π(t; x 2, g )) constant for all t R. Proof. Case (i): t n. It follows from Lemma 2.4(3) that there exists a T > 0 and an integer N 1 > 0 such that σ(π(t; x 1, g) π(t; x 2, g)) = N 1 (4.1) for all t T. Let t 0 R be such that π(t 0 ; x 1, g ) π(t 0 ; x 2, g ) Λ. Note that Π tn (x i, g) (x i, g )(i = 1, 2) and the continuity of σ, one has σ(π(t n +t 0 ; x 1, g) π(t n +t 0 ; x 2, g)) = σ(π(t 0 ; x 1, g ) π(t 0 ; x 2, g )) for all n sufficiently large. Consequently, σ(π(t 0 ; x 1, g ) π(t 0 ; x 2, g )) = N 1 from (4.1). By Lemma 2.4 (a),(b) and the arbitrariness of t 0, we have σ(π(t; x 1, g ) π(t; x 2, g )) = N 1 for all t R. Case (ii): t n. It follows from Lemma 2.4(4) that there exists a T > 0 and an integer N 2 > 0 such that σ(π(t; x 1, g) π(t; x 2, g)) = N 2 13

15 for all t T. The remainings are the same as above. Lemma 4.2. Let E 1, E 2 R n H(f) be two minimal sets of Π. Then there exists an integer N N such that for any g H(f) and any (x i, g) E i P 1 (g) (i = 1, 2), one has σ(π( ; x 1, g) π( ; x 2, g)) = N. Proof. Fix any g H(f) and any (x i, g) E i P 1 (g) (i = 1, 2), we first claim that there is an integer N > 0 such that σ(π(t; x 1, g) π(t; x 2, g)) = N for all t R. Indeed, it follows from Lemma 2.4 that there is a T > 0 and N 1, N 2 N such that σ(π(t; x 1, g) π(t; x 2, g)) = N 1, for all t T. (4.2) and σ(π(t; x 1, g) π(t; x 2, g)) = N 2, for all t T. (4.3) Now choose (ˆx 2, g) E 2 P 1 (g). By the minimality of E 1, there is a sequence {t n }, t n as n, such that Π tn (x 1, g) (x 1, g) as t n. Without loss of generality, we assume that Π tn (ˆx 2, g) ( x 2, g) as t n. By virtue of Lemma 4.1, there is an integer N > 0 such that σ(π(t; x 1, g) π(t; x 2, g)) = N, for all t R. (4.4) Note that (x 2, g), ( x 2, g) E 2 P 1 (g). then Theorem 3.1(a) implies that (x 2, g), ( x 2, g) forms a two sided proximal pair. Therefore, by (4.2)-(4.4) and the continuity of σ, one has N 1 = N = N 2. That is, σ(π(t; x 1, g) π(t; x 2, g)) = N, for all t T. So it follows from Lemma 2.4(2) that our claim is proved. Next we shall prove that the integer N is actually independent of g H(f) and (x i, g) E i P 1 (g)(i = 1, 2). To end this, for given g H(f), take any (x i, g), (ˆx i, g) E i P 1 (g) (i = 1, 2). By the claim above, there are N 1, N 2 N such that σ(π(t; x 1, g) π(t; x 2, g)) = N 1, for all t R, 14

16 and σ(π(t; ˆx 1, g) π(t; ˆx 2, g)) = N 2, for all t R. By virtue of Theorem 3.1(a), (x i, g), (ˆx i, g) forms a two sided proximal pair (i = 1, 2). It then follows from the continuity of σ that N 1 = σ(π(t; x 1, g) π(t; x 2, g)) = σ(π(t; x 1, g) π(t; ˆx 2, g)) = σ(π(t; ˆx 1, g) π(t; ˆx 2, g)) = N 2. Finally, take any g, ĝ H(f), and (x i, g) E i P 1 (g), (ˆx i, ĝ) E i P 1 (ĝ) (i = 1, 2). By the minimality of E i (i = 1, 2), there exist ( x i, g) E i P 1 (g) and a sequence {t n } with t n such that Π tn ( x i, g) (ˆx i, ĝ), (i = 1, 2) as n. By the arguments in the previous paragraph and in the proof of Lemma 4.1, we have N = σ(π(t; x 1, g) π(t; x 2, g)) = σ(π(t; x 1, g) π(t; x 2, g)) = σ(π(t; ˆx 1, ĝ) π(t; ˆx 2, ĝ)), for all t R. Thus we have proved that N is independent of g H(f) and (x i, g) E i P 1 (g)(i = 1, 2), which completes the proof of the Lemma. Lemma 4.3. Let E 1, E 2 R n H(f) be two minimal sets of Π. For each g H(f), define m i (g) := min { (x) 1 : (x, g) E i P 1 (g) }, M i (g) := max { (x) 1 : (x, g) E i P 1 (g) }. (4.5) Then E 1, E 2 are separated in the following sense: (a) [m 1 (g), M 1 (g)] [m 2 (g), M 2 (g)] = for all g H(f); (b) Without loss of generality, assume that m 1 (g 0 ) > M 2 (g 0 ) for some g 0 H(f), then there exists δ > 0 such that m 1 (g) > M 2 (g) + δ for all g H(f). Proof. We first claim that (a) holds for some g 0 H(f). Otherwise, one has m 2 (g) M 1 (g) and m 1 (g) M 2 (g) for all g H(f). Given g, g H(f), let (x 1, g) E 1 P 1 (g) be such that (x 1 ) 1 = m 1 (g) and (x 2, g) E 2 P 1 (g) be such that (x 2 ) 1 = M 2 (g). It follows from Lemma 15

17 4.2 that σ(π(t; x 1, g) π(t; x 2, g)) = constant, for all t R. By the definition of σ, we have (π(t; x 1, g)) 1 (π(t; x 2, g)) 1 0 for all t R. Note that (x 1 ) 1 = m 1 (g) M 2 (g) = (x 2 ) 1. Then (π(t; x 1, g)) 1 < (π(t; x 2, g)) 1, for all t R. (4.6) By the minimality of E 1, there is a sequence {t n }, t n, such that Π tn (x 1, g) (x 1, g ) as n, where (x 1, g ) E 1 with (x 1 ) 1 = M 1 (g ). Without loss of generality, we can aslo assume that Π tn (x 2, g) (x 2, g ) as n. By (4.6), M 1 (g ) = (x 1 ) 1 (x 2 ) 1 M 2 (g ). Moverover, it again follows from Lemma 4.2 that M 1 (g ) M 2 (g ), which implies that M 1 (g ) < M 2 (g ). However, similarly as above, one can also obtain that M 2 (g ) < M 1 (g ), a contradiction. Thus we have proved the claim. Now we can assume without loss of generality that there is a g 0 H(f) such that [m 1 (g 0 ), M 1 (g 0 )] < [m 2 (g 0 ), M 2 (g 0 )] (4.7) Suppose that there exists some g H(f) such that m 2 (g ) M 1 (g ). Then choose (x, g ) E 2 P 1 (g ) with (x ) 1 = m 2 (g ), and choose (y, g ) E 1 P 1 (g ) with (y ) 1 = M 1 (g ). By the minimality of E 2, we can find a sequence {t n }, t n, such that Π tn (x, g ) (x, g 0 ) as n, and (x ) 1 = m 2 (g 0 ). Without loss of generality, we assume that Π tn (y, g ) (y, g 0 ) as n. By the same arguments in the previous paragraph, one has m 2 (g 0 ) = (x ) 1 (y ) 1 M 1 (g 0 ), which contradicts (4.7). Therefore, m 2 (g) > M 1 (g) for all g H(f). We have proved (a). (b) can be easily obtained by the compactness of E 1, E 2. Theorem 4.1. Let (x 0, g 0 ) R n H(f) be such that its orbit Π t (x 0, g 0 )(t 0) is bounded. Then its ω-limit set ω(x 0, g 0 ) contains at most two minimal sets. Proof. Suppose that ω(x 0, g 0 ) contains three minimal sets E i, i = 1, 2, 3. Define A i (g) = { (x) 1 : (x, g) E i P 1 (g) } 16

18 and m i (g) = min A i (g), M i (g) = max A i (g). for all g H(f) and i = 1, 2, 3. By virtue of Lemma 4.3, we can assume without loss of generality that there is a δ > 0 such that M 1 (g) + δ m 2 (g) M 2 (g) < M 2 (g) + δ m 3 (g), (4.8) for all g H(f). Now choose (x i, g 0 ) E i P 1 (g 0 ), i = 1, 2, 3, and we consider (x 0, g 0 ) and (x 2, g 0 ). It follows from Lemma 2.4(3) that there is a T > 0 such that σ(π(t; x 0, g 0 ) π(t; x 2, g 0 )) = constant, for all t T. Then, by the continuity of σ, we can assume without loss of generality that (π(t; x 0, g 0 )) 1 < (π(t; x 2, g 0 )) 1 for all t T. Note that E 3 ω(x 0, g 0 ), so there is a sequence {t n }, t n, such that (π(t n ; x 0, g 0 )) 1 m 3 (g ) as n. Let (π(t n ; x 2, g 0 )) 1 β(g ) with β(g ) [m 2 (g ), M 2 (g )]. Consequently, m 3 (g ) β(g ) M 2 (g ), contradicting (4.8). This completes the proof. As mentioned in the introduction, even if f in (1.1) with (F2) is uniformly almost periodic, one can not always expect an ω-limit set ω(x 0, f) to be minimal, or contains only one minimal set, or a 1-cover of H(f) (if it is minimal). We now discuss some situation in which ω(x 0, f) can be a 1-cover of H(f). Now we assume that f in (1.1) with (F2) is uniformly almost periodic. We recall that the forward orbit Π t (x 0, f) of (1.3) is said to be uniformly stable if for any ε > 0, there is a δ > 0 such that if π(τ; x 0, f) π(τ; x, f) < δ(ε) for some (x, f) R n H(f), and some τ R, then π(t + τ; x 0, f) π(t + τ; x, f) < ε for all t 0. We have Corollary 4.1. (1) If the ω-limit set ω(x 0, f) is distal, then ω(x 0, f) is a 1-cover of H(f). (2) If the orbit Π t (x 0, f)(t 0) is bounded and uniformly stable. Then ω(x 0, f) is a 1-cover of H(f). 17

19 Proof. (1) It follows from Theorem 4.1 that ω(x 0, f) can be written as ω(x 0, f) = E 1 E 2 E 12, where E i (i = 1, 2) are minimal sets. Suppose that E 1 E 2, since ω(x 0, f) is connected, then E 12. So, for any (y, g) E 12, ω(y, g) (E 1 E 2 ) (otherwise, there will be at least three minimal set in ω(x 0, f), a contradiction). Note that ω(x 0, f) is distal, this is impossible. Hence E 1 = E 2, which implies that ω(x 0, f) = E 1 E 12. Similarly as above, one can prove that E 12 =. Consequently, ω(x 0, f) = E 1, which now is a distal and minimal set. Then, by Theorem 3.1, we have ω(x 0, f) is a 1-cover of H(f). (2) By [16], ω(x 0, f) is minimal and distal. Then ω(x 0, f) is a 1-cover of H(f). References [1] S. B. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math. 390(1988), [2] B. Fiedler and T. Gedeon, A Lyapunov function for tridiagonal competitive-cooperative systems, SIAM J. Math. Anal. 30(1999), [3] A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Math., Vol. 840, Springer-Verlag, Berlin, [4] H. I. Freedman and H. L. Smith, Tridiagonal competitive-cooperative Kolmogorov systems, Differential Equations Dynam. Systems 3(1995), [5] M. Gyllenberg and Y. Wang, Periodic tridiagonal systems modeling competitive-cooperative ecological interactions, Discrete Contin. Dyn. Syst. B, 5(2005), [6] G. Hetzer and W. Shen, Convergence in almost periodic competition diffusion systems, J. Math. Anal. Appl., 262(2001), [7] G. Hetzer and W. Shen, Uniform persistence, coexistence, and extinction in almost periodic/nonautonomous competition diffusion systems, SIAM J. Math. Anal. 34(2002),

20 [8] M. W. Hirsch, Stability and convergence in strongly monotone dynamical systems. J. reine angew. Math. 383 (1988), [9] R. A. Johnson, A linear almost periodic equations with an almost automorphic solution, Proc. Amer. Math. Soc. 82(1981). [10] H. Matano, Strongly comparison principle in nonlinear parabolic equations : in Nonlinear parabolic equations: qualitative properties of solutions (L. Boccardo and A. Tesei. eds.), Pitman Res. Notes in Math. 149 (Longman Scientific and Technical, 1987.), [11] H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation., J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29(1982), [12] K. Nickel, Gestattaussagen über Lösungen parabolischer Differentialgleichungern, J. Reine Angew. Math., 211(1962), [13] P. Poláčik, Convergence in smooth strongly monotone flows defined by semilinear parabolic equations, J. Diff. Eqns. 79 (1989), [14] P. Poláčik and I. Tereščák, Convergence to cycles as a typical asymptotic behavior in smoothly monotone discrete-time dynamical systems, Arch. Rat. Mech. Anal. 116 (1991), [15] R. J. Sacker and G. Sell, Lifting properties in skew-product flows with applications to differential equations, Mem. Amer. Math. Soc. 11 (1977). [16] G. Sell, Topological Dynamics and Ordinary Differential Equations, Vn Norstand Reinhold Company, [17] W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Memoirs Amer. Math. Soc. 136 (1998). [18] W. Shen and Y. Yi, Dynamics of almost periodic scalar parabolic equations, J. Diff. Eqns. 122(1995),

21 [19] W. Shen and Y. Yi, Asymptitic almost periodicity of scalar parabolic equations with almost periodic time dependence, J. Diff. Eqns. 122(1995), [20] W. Shen and Y. Yi, On minimal sets of scalar parabolic equations with skew-product structures, Trans. Amer. Math. Soc. 347 (1995), [21] W. Shen and Y. Wang, Carrying Simplices in Nonautonomous and Random Competitive Kolmogorov Systems, preprint, [22] J. Smillie, Competitive and cooperative tridiagonal systems of differential equations, SIAM J. Math. Anal. 15(1984), [23] H. L. Smith, Periodic tridiagonal competitive and cooperative systems of differential equations. SIAM J. Math. Anal. 22(1991), [24] H. L. Smith and H. R. Theime, Quasiconvergence and stability for strongly order-preserving semiflows, SIAM, J. Math. Anal. 21 (1990), [25] H. L. Smith and H. R. Theime, Convergence for strongly order-preserving semiflows, SIAM, J. Math. Anal. 22 (1991), [26] I. Tereščák, Dynamics of C 1 smooth strongly monotone discrete-time dynamical systems, preprint, [27] W. A. Veech, Almost automorphic functions on groups, Amer. J. Math. 87 (1965), [28] X.-Q. Zhao, Convergence in asymptotically periodic tridiagonal competitive-cooperative systems, Canad. Appl. Math. Quart. 6(1998),