Convergence in almost periodic cooperative systems with a first integral
Authors:
Wenxian Shen and XiaoQiang Zhao
Journal:
Proc. Amer. Math. Soc. 133 (2005), 203212
MSC (2000):
Primary 34C12, 34C27, 37B55
Published electronically:
June 18, 2004
MathSciNet review:
2085171
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: This paper is to investigate the asymptotic dynamics in almost periodic cooperative systems with a first integral. By appealing to the theory of skewproduct 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), 13311339.
 1.
O. Arino, Monotone semiflows which have a monotone first integral, in ``Delay Differential Equations and Dynamical Systems'', Lecture Notes in Math. 1475, SpringerVerlag, Berlin/Heidelberg, 1991, pp. 6475. MR 93a:34081
 2.
XuYan
Chen and Hiroshi
Matano, Convergence, asymptotic periodicity, and finitepoint
blowup in onedimensional semilinear heat equations, J. Differential
Equations 78 (1989), no. 1, 160–190. MR 986159
(90e:35018), http://dx.doi.org/10.1016/00220396(89)900818
 3.
A.
M. Fink, Almost periodic differential equations, Lecture Notes
in Mathematics, Vol. 377, SpringerVerlag, BerlinNew York, 1974. MR 0460799
(57 #792)
 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
(2003m:37143), http://dx.doi.org/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 (89c:58108), http://dx.doi.org/10.1515/crll.1988.383.1
 6.
Jifa
Jiang, Type 𝐾monotone systems with an orderincreasing
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
(97g:34062)
 7.
JiFa
Jiang, Periodic monotone systems with an invariant function,
SIAM J. Math. Anal. 27 (1996), no. 6,
1738–1744. MR 1416516
(98h:34089), http://dx.doi.org/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 (88e:34093), http://dx.doi.org/10.1137/0518049
 9.
F.
Nakajima, Periodic time dependent grosssubstitute systems,
SIAM J. Appl. Math. 36 (1979), no. 3, 421–427.
MR 531605
(80c:93045), http://dx.doi.org/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
(90f:58025), http://dx.doi.org/10.1016/00220396(89)901150
 11.
Robert
J. Sacker and George
R. Sell, Lifting properties in skewproduct flows with applications
to differential equations, Mem. Amer. Math. Soc. 11
(1977), no. 190, iv+67. MR 0448325
(56 #6632)
 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
(56 #1283)
 13.
George
R. Sell and Fumio
Nakajima, Almost periodic grosssubstitute dynamical systems,
Tôhoku Math. J. (2) 32 (1980), no. 2,
255–263. MR
580280 (81j:34075), http://dx.doi.org/10.2748/tmj/1178229641
 14.
Wenxian
Shen and Yingfei
Yi, Almost automorphic and almost periodic dynamics in skewproduct
semiflows, Mem. Amer. Math. Soc. 136 (1998),
no. 647, x+93. MR 1445493
(99d:34088), http://dx.doi.org/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
(96c:34002)
 16.
Peter
Takáč, Asymptotic behavior of strongly monotone
timeperiodic dynamical processes with symmetry, J. Differential
Equations 100 (1992), no. 2, 355–378. MR 1194815
(94d:47060), http://dx.doi.org/10.1016/00220396(92)901198
 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
(94g:34067), http://dx.doi.org/10.1137/0524076
 18.
X.Q. Zhao, Global attractivity in monotone and subhomogeneous almost periodic systems, J. Differential Equations, 187 (2003), 494509.
 19.
X.Q. Zhao, ``Dynamical Systems in Population Biology'', CMS books in Mathematics, Vol. 16, SpringerVerlag, New York, 2003.
 1.
 O. Arino, Monotone semiflows which have a monotone first integral, in ``Delay Differential Equations and Dynamical Systems'', Lecture Notes in Math. 1475, SpringerVerlag, Berlin/Heidelberg, 1991, pp. 6475. MR 93a:34081
 2.
 X.Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finitepoint blowup in onedimensional semilinear heat equations, J. Diff. Eqs., 78 (1989), 160190. MR 90e:35018
 3.
 A. M. Fink, ``Almost Periodic Differential Equations'', Lecture Notes in Mathematics 377, SpringerVerlag, Berlin/Heidelberg/New York, 1974. MR 57:792
 4.
 G. Hetzer and W. Shen, Uniform persistence, coexistence, and extinction in almost periodic/nonautonomous competition diffusion systems, SIAM J. Math. Anal., 34 (2002), 204227. MR 2003m:37143
 5.
 M. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math., 383 (1988), 153. MR 89c:58108
 6.
 J.F. Jiang, Type monotone systems with an orderincreasing invariant function, Chinese Ann. Math., Ser. B, 17 (1996), 335342. MR 97g:34062
 7.
 J.F. Jiang, Periodic monotone systems with an invariant function, SIAM J. Math. Anal., 27 (1996), 17381744. MR 98h:34089
 8.
 J. Mierczynski, Strictly cooperative systems with a first integral, SIAM J. Math. Anal., 18 (1987), 642646. MR 88e:34093
 9.
 F. Nakajima, Periodic timedependent grosssubstitute systems, SIAM J. Appl. Math., 36 (1979), 421427. MR 80c:93045
 10.
 P. Polacik, Convergence in smooth strongly monotone flows defined by semilinear parabolic equations, J. Diff. Eqs., 79 (1989), 89110. MR 90f:58025
 11.
 R. J. Sacker and G. R. Sell, ``Lifting Properties in SkewProduct Flows with Applications to Differential Equations'', Memoirs Amer. Math. Soc. , No. 190, Vol. 11, Providence, R.I., 1977. MR 56:6632
 12.
 G. Sell, ``Topological Dynamics and Ordinary Differential Equations'', Van Nostrand Reinhold, London, 1971. MR 56:1283
 13.
 G. R. Sell and F. Nakajima, Almost periodic timedependent grosssubstitute dynamical systems, Tohoku Math. J., 32 (1980), 255263. MR 81j:34075
 14.
 W. Shen and Y. Yi, ``Almost Automorphic and Almost Periodic Dynamics in Skewproduct Semiflows", Memoirs Amer. Math. Soc., No. 647, Vol. 136, Providence, R.I., 1998. MR 99d:34088
 15.
 H. L. Smith, ``Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Systems'', Mathematical Surveys and Monographs, Vol. 41, Amer. Math. Soc., Providence, RI, 1995. MR 96c:34002
 16.
 P. Takac, Asymptotic behavior of strongly monotone timeperiodic dynamical processes with symmetry, J. Diff. Eqs., 100 (1992), 355378. MR 94d:47060
 17.
 B. Tang, Y. Kuang and H. Smith, Strictly nonautonomous cooperative system with a first integral, SIAM J. Math. Anal., 24 (1993), 13311339. MR 94g:34067
 18.
 X.Q. Zhao, Global attractivity in monotone and subhomogeneous almost periodic systems, J. Differential Equations, 187 (2003), 494509.
 19.
 X.Q. Zhao, ``Dynamical Systems in Population Biology'', CMS books in Mathematics, Vol. 16, SpringerVerlag, New York, 2003.
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
34C12,
34C27,
37B55
Retrieve articles in all journals
with MSC (2000):
34C12,
34C27,
37B55
Additional Information
Wenxian Shen
Affiliation:
Department of Mathematics, Auburn University, Auburn, Alabama 36849
Email:
ws@math.auburn.edu
XiaoQiang 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/S0002993904075562
PII:
S 00029939(04)075562
Keywords:
Cooperative systems,
first integral,
almost periodic solutions,
skewproduct 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 DMS0103381
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
