Convergence in almost periodic cooperative systems with a first integral

Authors:
Wenxian Shen and Xiao-Qiang Zhao

Journal:
Proc. Amer. Math. Soc. **133** (2005), 203-212

MSC (2000):
Primary 34C12, 34C27, 37B55

Published electronically:
June 18, 2004

MathSciNet review:
2085171

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Abstract: This paper is to investigate the asymptotic dynamics in almost periodic cooperative systems with a first integral. By appealing to the theory of skew-product semiflows we establish the asymptotic almost periodicity of bounded solutions to such systems, which extends the existing convergence results for time independent and periodic cooperative systems with a first integral and proves a conjecture of B. Tang, Y. Kuang and H. Smith in *SIAM J. Math. Anal.*, 24 (1993), 1331-1339.

**1.**Ovide Arino,*Monotone semi-flows which have a monotone first integral*, Delay differential equations and dynamical systems (Claremont, CA, 1990), Lecture Notes in Math., vol. 1475, Springer, Berlin, 1991, pp. 64–75. MR**1132019**, 10.1007/BFb0083480**2.**Xu-Yan Chen and Hiroshi Matano,*Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations*, J. Differential Equations**78**(1989), no. 1, 160–190. MR**986159**, 10.1016/0022-0396(89)90081-8**3.**A. M. Fink,*Almost periodic differential equations*, Lecture Notes in Mathematics, Vol. 377, Springer-Verlag, Berlin-New York, 1974. MR**0460799****4.**Georg Hetzer and Wenxian Shen,*Uniform persistence, coexistence, and extinction in almost periodic/nonautonomous competition diffusion systems*, SIAM J. Math. Anal.**34**(2002), no. 1, 204–227. MR**1950832**, 10.1137/S0036141001390695**5.**Morris W. Hirsch,*Stability and convergence in strongly monotone dynamical systems*, J. Reine Angew. Math.**383**(1988), 1–53. MR**921986**, 10.1515/crll.1988.383.1**6.**Jifa Jiang,*Type 𝐾-monotone systems with an order-increasing invariant function*, Chinese Ann. Math. Ser. B**17**(1996), no. 3, 335–342. A Chinese summary appears in Chinese Ann. Math. Ser. A 17 (1996), no. 4, 516. MR**1415261****7.**Ji-Fa Jiang,*Periodic monotone systems with an invariant function*, SIAM J. Math. Anal.**27**(1996), no. 6, 1738–1744. MR**1416516**, 10.1137/S003614109326063X**8.**Janusz Mierczyński,*Strictly cooperative systems with a first integral*, SIAM J. Math. Anal.**18**(1987), no. 3, 642–646. MR**883558**, 10.1137/0518049**9.**F. Nakajima,*Periodic time dependent gross-substitute systems*, SIAM J. Appl. Math.**36**(1979), no. 3, 421–427. MR**531605**, 10.1137/0136032**10.**Peter Poláčik,*Convergence in smooth strongly monotone flows defined by semilinear parabolic equations*, J. Differential Equations**79**(1989), no. 1, 89–110. MR**997611**, 10.1016/0022-0396(89)90115-0**11.**Robert J. Sacker and George R. Sell,*Lifting properties in skew-product flows with applications to differential equations*, Mem. Amer. Math. Soc.**11**(1977), no. 190, iv+67. MR**0448325****12.**George R. Sell,*Topological dynamics and ordinary differential equations*, Van Nostrand Reinhold Co., London, 1971. Van Nostrand Reinhold Mathematical Studies, No. 33. MR**0442908****13.**George R. Sell and Fumio Nakajima,*Almost periodic gross-substitute dynamical systems*, Tôhoku Math. J. (2)**32**(1980), no. 2, 255–263. MR**580280**, 10.2748/tmj/1178229641**14.**Wenxian Shen and Yingfei Yi,*Almost automorphic and almost periodic dynamics in skew-product semiflows*, Mem. Amer. Math. Soc.**136**(1998), no. 647, x+93. MR**1445493**, 10.1090/memo/0647**15.**Hal L. Smith,*Monotone dynamical systems*, Mathematical Surveys and Monographs, vol. 41, American Mathematical Society, Providence, RI, 1995. An introduction to the theory of competitive and cooperative systems. MR**1319817****16.**Peter Takáč,*Asymptotic behavior of strongly monotone time-periodic dynamical processes with symmetry*, J. Differential Equations**100**(1992), no. 2, 355–378. MR**1194815**, 10.1016/0022-0396(92)90119-8**17.**Bao Rong Tang, Yang Kuang, and Hal Smith,*Strictly nonautonomous cooperative system with a first integral*, SIAM J. Math. Anal.**24**(1993), no. 5, 1331–1339. MR**1234019**, 10.1137/0524076**18.**X.-Q. Zhao, Global attractivity in monotone and subhomogeneous almost periodic systems,*J. Differential Equations*,**187**(2003), 494-509.**19.**X.-Q. Zhao, ``Dynamical Systems in Population Biology'', CMS books in Mathematics, Vol. 16, Springer-Verlag, New York, 2003.

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Additional Information

**Wenxian Shen**

Affiliation:
Department of Mathematics, Auburn University, Auburn, Alabama 36849

Email:
ws@math.auburn.edu

**Xiao-Qiang Zhao**

Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Newfoundland, Canada A1C 5S7

Email:
xzhao@math.mun.ca

DOI:
http://dx.doi.org/10.1090/S0002-9939-04-07556-2

Keywords:
Cooperative systems,
first integral,
almost periodic solutions,
skew-product semiflows

Received by editor(s):
June 17, 2003

Received by editor(s) in revised form:
September 24, 2003

Published electronically:
June 18, 2004

Additional Notes:
The first author’s research was supported in part by NSF grant DMS-0103381

The second author’s research was supported in part by the NSERC of Canada

Communicated by:
Carmen C. Chicone

Article copyright:
© Copyright 2004
American Mathematical Society