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

Li, Yongkun; Kuang, Yang

doi:10.1090/S0002-9939-01-06444-9

Periodic solutions in periodic state-dependent delay equations and population models
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by Yongkun Li and Yang Kuang PDF

Proc. Amer. Math. Soc. 130 (2002), 1345-1353 Request permission

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.

References

H. I. Freedman and Jian Hong Wu, Periodic solutions of single-species models with periodic delay, SIAM J. Math. Anal. 23 (1992), no. 3, 689–701. MR 1158828, DOI 10.1137/0523035
Robert E. Gaines and Jean L. Mawhin, Coincidence degree, and nonlinear differential equations, Lecture Notes in Mathematics, Vol. 568, Springer-Verlag, Berlin-New York, 1977. MR 0637067, DOI 10.1007/BFb0089537
K. Gopalsamy, M. R. S. Kulenović, and G. Ladas, Environmental periodicity and time delays in a “food-limited” population model, J. Math. Anal. Appl. 147 (1990), no. 2, 545–555. MR 1050226, DOI 10.1016/0022-247X(90)90369-Q
Yang Kuang, Delay differential equations with applications in population dynamics, Mathematics in Science and Engineering, vol. 191, Academic Press, Inc., Boston, MA, 1993. MR 1218880
Yongkun Li, Periodic solutions of a periodic delay predator-prey system, Proc. Amer. Math. Soc. 127 (1999), no. 5, 1331–1335. MR 1646198, DOI 10.1090/S0002-9939-99-05210-7
Yongkun Li and Yang Kuang, Periodic solutions of periodic delay Lotka-Volterra equations and systems, J. Math. Anal. Appl. 255 (2001), no. 1, 260–280. MR 1813821, DOI 10.1006/jmaa.2000.7248
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.
R. M. May (1974), Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton.
H. L. Smith and Y. Kuang, Periodic solutions of differential delay equations with threshold-type delays, Oscillation and dynamics in delay equations (San Francisco, CA, 1991) Contemp. Math., vol. 129, Amer. Math. Soc., Providence, RI, 1992, pp. 153–176. MR 1174140, DOI 10.1090/conm/129/1174140
Baorong Tang and Yang Kuang, Existence, uniqueness and asymptotic stability of periodic solutions of periodic functional-differential systems, Tohoku Math. J. (2) 49 (1997), no. 2, 217–239. MR 1447183, DOI 10.2748/tmj/1178225148
B. G. Zhang and K. Gopalsamy, Global attractivity and oscillations in a periodic delay-logistic equation, J. Math. Anal. Appl. 150 (1990), no. 1, 274–283. MR 1059588, DOI 10.1016/0022-247X(90)90213-Y
Tao Zhao, Yang Kuang, and H. L. Smith, Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems, Nonlinear Anal. 28 (1997), no. 8, 1373–1394. MR 1428657, DOI 10.1016/0362-546X(95)00230-S