On positive periodic solutions of Lotka-Volterra competition systems with deviating arguments

Authors:
Xianhua Tang and Xingfu Zou

Journal:
Proc. Amer. Math. Soc. **134** (2006), 2967-2974

MSC (2000):
Primary 34K13; Secondary 34K20, 92D25

DOI:
https://doi.org/10.1090/S0002-9939-06-08320-1

Published electronically:
May 9, 2006

MathSciNet review:
2231621

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: By using Krasnoselskii's fixed point theorem, we prove that the following periodic species Lotka-Volterra competition system with multiple deviating arguments

**[1]**C. Alvarez, A. C. Lazer,*An application of topological degree to the periodic competiting species model,*J. Austral. Math. Soc. Ser. B 28(1986), 202-219. MR**0862570 (87k:34062)****[2]**S. Ahmad,*On the nonautonomous Lotka-Volterra competition equations,*Proc. Amer. Math. Soc. 117(1993), 199-204. MR**1143013 (93c:34109)****[3]**A. Battaaz, F. Zanolin,*Coexistence states for periodic competition Kolmogorov systems,*J. Math. Anal. Appl. 219(1998), 179-199. MR**1606377 (98m:34086)****[4]**Y. Chen, Z. Zhou,*Stable periodic solution of a discrete periodic Lotka-Volterra competition system,*J. Math. Anal. Appl. 277(2003), 358-366. MR**1954481 (2004k:39032)****[5]**J. M. Cushing,*Two species competition in a periodic environment,*J. Math. Biol. 10(1980), 385-400. MR**0602256 (82c:92017)****[6]**M. Fan, K. Wang,*Global Periodic Solutions of a Generalized Species Gilpin-Ayala Competition Model,*Computers Math. Applic. 40(2000), 1141-1151. MR**1784658 (2001i:92043)****[7]**M. Fan, K. Wang, D. Q. Jiang,*Existence and global attractivity of positive periodic solutions of periodic species Lotka-Volterra competition systems with several deviating arguments,*

Math. Biosci. 160(1999), 47-61. MR**1704338 (2000f:92016)****[8]**H. I. Freedman, P. Waltman,*Persistence in a model of three competitive populations,*Math. Biosci. 73(1985), 89-101. MR**0779763 (86i:92038)****[9]**K. Gopalsamy,*Global asymptotical stability in a periodic Lotka-Volterra system,*J. Austral. Math. Soc. Ser. B 24(1982), 160-.**[10]**K. Gopalsamy,*Global asymptotical stability in a periodic Lotka-Volterra system,*J. Austral. Math. Soc. Ser. B 29(1985), 66-72.**[11]**K. Gopalsamy,*Stability and Oscillation in Delay Differential Equations of Population Dynamics,*Kluwer Academic Publishers, Dordrecht, 1992. MR**1163190 (93c:34150)****[12]**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)****[14]**P. Korman,*Some new results on the periodic competition model,*J. Math. Anal. Appl. 171(1992), 131-138. MR**1192498 (93j:92033)****[15]**Y. Kuang,*Delay Differential Equations with Applications in Population Dynamics,*Academic Press, Boston, 1993. MR**1218880 (94f:34001)****[16]**Y. K. Li,*Periodic solutions of species competition system with delays,*J. Biomath. 12(1997), 1-12. MR**1460907 (99f:92021)****[17]**Y. K. Li,*On a periodic delay logistic type population model,*Ann. of Diff. Eqs. 14(1998), 29-36. MR**1633664 (99d:34133)****[18]**Y. K. Li,*Periodic solutions for delay Lotka-Volterra competition systems,*J. Math. Anal. Appl. 246(2000), 230-244. MR**1761160 (2001b:34132)****[19]**Y. K. Li, Y. Kuang,*Periodic solutions of periodic delay Lotka-Volterra equations and systems,*J. Math. Anal. Appl. 255(2001), 260-280. MR**1813821 (2001k:34133)****[20]**H. L. Roydin,*Real Analysis,*

Macmillan Publishing Company, New York, 1988.MR**1013117 (90g:00004)****[21]**A. Shibata, N. Saito,*Time delays and chaos in two competition system,*Math. Biosci. 51(1980), 199-211. MR**0587228 (81m:92054)****[22]**H. L. Smith,*Periodic solutions of periodic competitive and cooperative systems,*SIAM J. Math. Anal. 17(1986), 1289-1318. MR**0860914 (87m:34057)****[23]**H. L. Smith,*Periodic competitive differential and the descrete dynamics of a competitive map,*J. Different. Eq. 64(1986), 165-194. MR**0851910 (87k:92027)****[24]**A. Trieo, C. Alrarez,*A different consideration about the globally asymptotically stable solution of the periodic competing species problem,*J. Math. Anal. Appl. 159(1991), 44-50. MR**1119420 (93d:34080)****[25]**X. H. Tang and X. Zou,*3/2-type criteria for global attractivity of Lotka-Volterra competition system without instantaneous negative feedback,*J. Differential Equations, 186(2002), 420-439. MR**1942216 (2003k:34138)****[26]**X. H. Tang and X. Zou,*Global attractivity of non-autonomous Lotka-Volterra competition system without instantaneous negative feedbacks,*J. Differential Equations, 192(2003), 502-535. MR**1990850 (2004e:34116)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
34K13,
34K20,
92D25

Retrieve articles in all journals with MSC (2000): 34K13, 34K20, 92D25

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:
https://doi.org/10.1090/S0002-9939-06-08320-1

Keywords:
Positive periodic solution,
Lotka-Volterra 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 Start-Up 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.