Periodic solutions in periodic state-dependent delay equations and population models

Authors:
Yongkun Li and Yang Kuang

Journal:
Proc. Amer. Math. Soc. **130** (2002), 1345-1353

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

DOI:
https://doi.org/10.1090/S0002-9939-01-06444-9

Published electronically:
December 27, 2001

MathSciNet review:
1879956

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Sufficient and realistic conditions are obtained for the existence of positive periodic solutions in periodic equations with state-dependent delay. The method involves the application of the coincidence degree theorem and estimations of uniform upper bounds on solutions. Applications of these results to some population models are presented. These application results indicate that seasonal effects on population models often lead to synchronous solutions. In addition, we may conclude that when both seasonality and time delay are present and deserve consideration, the seasonality is often the generating force for the often observed oscillatory behavior in population densities.

**1.**H. I. Freedman and J. Wu(1992), Periodic solutions of single-species models with periodic delay,*SIAM J. Math. Anal.*, 23, 689-701. MR**93e:92012****2.**R. E. Gaines and J. L. Mawhin(1977),*Coincidence Degree and Nonlinear Differential Equations*, Springer-Verlag, Berlin. MR**58:30551****3.**K. Gopalsamy, M. R. S. Kulenovic and G. Ladas(1990), Environmental periodicity and time delays in a ``food-limited'' population model,*J. Math. Anal. Appl.*147, 545-555. MR**91f:92020****4.**Y. Kuang(1993),*Delay Differential Equations with Applications in Population Dynamics*, Academic Press, Boston. MR**94f:34001****5.**Y. Li(1999), Periodic solutions of a periodic delay predator-prey system,*Proc. Amer. Math. Soc.*127, 1331-1335. MR**99i:34101****6.**Y. Li and Y. Kuang(2001a), Periodic solutions in periodic delay Lotka-Volterra equations and systems,*J. Math. Anal. Appl.*, 255, 260-280. MR**2001k:34133****7.**Y. Li and Y. Kuang(2001b), Periodic solutions in periodic delayed Gause-type predator-prey systems,*Proceeding of DYNAMIC SYSTEMS AND APPLICATIONS,*3, 375-382.**8.**R. M. May (1974),*Stability and Complexity in Model Ecosystems*, Princeton University Press, Princeton.**9.**H. L. Smith and Y. Kuang(1992), Periodic solutions of delay differential equations of threshold-type delays, in:*Oscillation and Dynamics in Delay Equations*, Graef and Hale, eds., 153-176, Contemporary Mathematics 129, AMS, Providence. MR**93j:34106****10.**B. R. Tang and Y. Kuang(1997), Existence, uniqueness and asymptotic stability of periodic solutions of periodic functional differential systems,*Tohoku Mathematical Journal*, 49, 217-239. MR**98g:34117****11.**B. G. Zhang and K. Gopalsamy(1990), Global attractivity and oscillations in a periodic delay-logistic equation,*J. Math. Anal. Appl.*150, 274-283. MR**91h:34120****12.**T. Zhao, Y. Kuang and H. L. Smith (1997), Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems,*Nonlinear Analysis, TMA*, 28, 1373-1394. MR**97m:34145**

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Additional Information

**Yongkun Li**

Affiliation:
Department of Mathematics, Yunnan University, Kunming, People’s Republic of China

**Yang Kuang**

Affiliation:
Department of Mathematics, Arizona State University, Tempe, Arizona 85287

Email:
kuang@asu.edu

DOI:
https://doi.org/10.1090/S0002-9939-01-06444-9

Keywords:
Coincidence degree,
periodic solution,
delay equation,
state-dependent delay,
population model

Received by editor(s):
July 1, 2000

Published electronically:
December 27, 2001

Additional Notes:
The second author’s research was partially supported by NSF Grant DMS-0077790

Communicated by:
Suncica Canic

Article copyright:
© Copyright 2001
American Mathematical Society