On positive periodic solutions of LotkaVolterra competition systems with deviating arguments
Authors:
Xianhua Tang and Xingfu Zou
Journal:
Proc. Amer. Math. Soc. 134 (2006), 29672974
MSC (2000):
Primary 34K13; Secondary 34K20, 92D25
Published electronically:
May 9, 2006
MathSciNet review:
2231621
Fulltext PDF Free Access
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Abstract: By using Krasnoselskii's fixed point theorem, we prove that the following periodic species LotkaVolterra competition system with multiple deviating arguments has at least one positive periodic solution provided that the corresponding system of linear equations has a positive solution, where and are periodic functions with Furthermore, when and , , are constants but , remain periodic, we show that the condition on is also necessary for to have at least one positive periodic solution.
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 C. Alvarez, A. C. Lazer, An application of topological degree to the periodic competiting species model, J. Austral. Math. Soc. Ser. B 28(1986), 202219. MR 0862570 (87k:34062)
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 [3]
 A. Battaaz, F. Zanolin, Coexistence states for periodic competition Kolmogorov systems, J. Math. Anal. Appl. 219(1998), 179199. MR 1606377 (98m:34086)
 [4]
 Y. Chen, Z. Zhou, Stable periodic solution of a discrete periodic LotkaVolterra competition system, J. Math. Anal. Appl. 277(2003), 358366. MR 1954481 (2004k:39032)
 [5]
 J. M. Cushing, Two species competition in a periodic environment, J. Math. Biol. 10(1980), 385400. MR 0602256 (82c:92017)
 [6]
 M. Fan, K. Wang, Global Periodic Solutions of a Generalized Species GilpinAyala Competition Model, Computers Math. Applic. 40(2000), 11411151. MR 1784658 (2001i:92043)
 [7]
 M. Fan, K. Wang, D. Q. Jiang, Existence and global attractivity of positive periodic solutions of periodic species LotkaVolterra competition systems with several deviating arguments,
Math. Biosci. 160(1999), 4761. MR 1704338 (2000f:92016)
 [8]
 H. I. Freedman, P. Waltman, Persistence in a model of three competitive populations, Math. Biosci. 73(1985), 89101. MR 0779763 (86i:92038)
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 K. Gopalsamy, Global asymptotical stability in a periodic LotkaVolterra system, J. Austral. Math. Soc. Ser. B 24(1982), 160.
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 K. Gopalsamy, Global asymptotical stability in a periodic LotkaVolterra system, J. Austral. Math. Soc. Ser. B 29(1985), 6672.
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 K. Gopalsamy, Stability and Oscillation in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, Dordrecht, 1992. MR 1163190 (93c:34150)
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 I. Györy, G. Ladas, Oscillation Theory of Delay Differential Equations, Oxford Science, Oxford, 1991. MR 1168471 (93m:34109)
 [13]
 M. A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964. MR 0181881 (31:6107)
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 P. Korman, Some new results on the periodic competition model, J. Math. Anal. Appl. 171(1992), 131138. MR 1192498 (93j:92033)
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 Y. K. Li, On a periodic delay logistic type population model, Ann. of Diff. Eqs. 14(1998), 2936. MR 1633664 (99d:34133)
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 Y. K. Li, Periodic solutions for delay LotkaVolterra competition systems, J. Math. Anal. Appl. 246(2000), 230244. MR 1761160 (2001b:34132)
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 Y. K. Li, Y. Kuang, Periodic solutions of periodic delay LotkaVolterra equations and systems, J. Math. Anal. Appl. 255(2001), 260280. MR 1813821 (2001k:34133)
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 H. L. Roydin, Real Analysis,
Macmillan Publishing Company, New York, 1988.MR 1013117 (90g:00004)
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 A. Shibata, N. Saito, Time delays and chaos in two competition system, Math. Biosci. 51(1980), 199211. MR 0587228 (81m:92054)
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 X. H. Tang and X. Zou, Global attractivity of nonautonomous LotkaVolterra competition system without instantaneous negative feedbacks, J. Differential Equations, 192(2003), 502535. MR 1990850 (2004e:34116)
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Additional Information
Xianhua Tang
Affiliation:
School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, People’s Republic of China
Email:
tangxh@mail.csu.edu.cn
Xingfu Zou
Affiliation:
Department of Applied Mathematics, University of Western Ontario, London, Ontario, Canada N6A 5B7
Email:
xzou@uwo.ca
DOI:
http://dx.doi.org/10.1090/S0002993906083201
PII:
S 00029939(06)083201
Keywords:
Positive periodic solution,
LotkaVolterra competition system
Received by editor(s):
August 13, 2004
Received by editor(s) in revised form:
April 29, 2005
Published electronically:
May 9, 2006
Additional Notes:
The first author was supported in part by NNSF of China (No. 10471153), and the second author was supported in part by the NSERC of Canada and by a Faculty of Science Dean’s StartUp Grant at the University of Western Ontario.
Communicated by:
Carmen C. Chicone
Article copyright:
© Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
