On the stability of sets for delayed Kolmogorov-type systems
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Abstract:
In this paper we consider Kolmogorov-type delay systems. Criteria on the uniform global asymptotic stability of sets are established for the above systems using Lyapunov functions and the Razumikhin technique.References
- Jacques Bélair and Michael C. Mackey, Consumer memory and price fluctuations in commodity markets: an integrodifferential model, J. Dynam. Differential Equations 1 (1989), no. 3, 299–325. MR 1010969, DOI 10.1007/BF01053930
- T. A. Burton, Uniform asymptotic stability in functional differential equations, Proc. Amer. Math. Soc. 68 (1978), no. 2, 195–199. MR 481371, DOI 10.1090/S0002-9939-1978-0481371-5
- M. M. A. El-Sheikh, Lipschitz stability criteria for a generalized delayed Kolmogorov model, J. Appl. Math. Comput. 10 (2002), no. 1-2, 75–81. MR 1922171, DOI 10.1007/BF02936207
- Teresa Faria, An asymptotic stability result for scalar delayed population models, Proc. Amer. Math. Soc. 132 (2004), no. 4, 1163–1169. MR 2045433, DOI 10.1090/S0002-9939-03-07237-X
- H. I. Freedman and A. A. Martynyuk, Boundedness criteria for solutions of perturbed Kolmogorov population models, Proceedings of the G. J. Butler Workshop in Mathematical Biology (Waterloo, ON, 1993), 1995, pp. 203–217. MR 1360032
- H. I. Freedman and Shi Gui Ruan, Uniform persistence in functional-differential equations, J. Differential Equations 115 (1995), no. 1, 173–192. MR 1308612, DOI 10.1006/jdeq.1995.1011
- Jack K. Hale and Sjoerd M. Verduyn Lunel, Introduction to functional-differential equations, Applied Mathematical Sciences, vol. 99, Springer-Verlag, New York, 1993. MR 1243878, DOI 10.1007/978-1-4612-4342-7
- V. Lakshmikantham, S. Leela, and A. A. Martynyuk, Stability analysis of nonlinear systems, Monographs and Textbooks in Pure and Applied Mathematics, vol. 125, Marcel Dekker, Inc., New York, 1989. MR 984861
- Bingwen Liu, Global stability of a class of non-autonomous delay differential systems, Proc. Amer. Math. Soc. 138 (2010), no. 3, 975–985. MR 2566564, DOI 10.1090/S0002-9939-09-10181-8
- James H. Liu, Uniform asymptotic stability via Liapunov-Razumikhin technique, Proc. Amer. Math. Soc. 123 (1995), no. 8, 2465–2471. MR 1257116, DOI 10.1090/S0002-9939-1995-1257116-8
- Shigui Ruan, Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays, Quart. Appl. Math. 59 (2001), no. 1, 159–173. MR 1811101, DOI 10.1090/qam/1811101
- Wenxian Shen and Yi Wang, Carrying simplices in nonautonomous and random competitive Kolmogorov systems, J. Differential Equations 245 (2008), no. 1, 1–29. MR 2422708, DOI 10.1016/j.jde.2008.03.024
- Wenxian Shen and Yingfei Yi, Convergence in almost periodic Fisher and Kolmogorov models, J. Math. Biol. 37 (1998), no. 1, 84–102. MR 1636648, DOI 10.1007/s002850050121
- Gani T. Stamov, Almost periodic solutions of impulsive differential equations, Lecture Notes in Mathematics, vol. 2047, Springer, Heidelberg, 2012. MR 2934087, DOI 10.1007/978-3-642-27546-3
- Ivanka Stamova, Stability analysis of impulsive functional differential equations, De Gruyter Expositions in Mathematics, vol. 52, Walter de Gruyter GmbH & Co. KG, Berlin, 2009. MR 2604930, DOI 10.1515/9783110221824
- Baorong Tang and Yang Kuang, Permanence in Kolmogorov-type systems of nonautonomous functional-differential equations, J. Math. Anal. Appl. 197 (1996), no. 2, 427–447. MR 1372189, DOI 10.1006/jmaa.1996.0030
- Zhidong Teng, Persistence and stability in general nonautonomous single-species Kolmogorov systems with delays, Nonlinear Anal. Real World Appl. 8 (2007), no. 1, 230–248. MR 2268081, DOI 10.1016/j.nonrwa.2005.08.003
- Taro Yoshizawa, Stability theory by Liapunov’s second method, Publications of the Mathematical Society of Japan, vol. 9, Mathematical Society of Japan, Tokyo, 1966. MR 0208086
Additional Information
- Ivanka M. Stamova
- Affiliation: Department of Mathematics, The University of Texas at San Antonio, One UTSA Circle, San Antonio, Texas 78249
- MR Author ID: 329335
- Email: ivanka.stamova@utsa.edu
- Gani Tr. Stamov
- Affiliation: Department of Mathematics, Technical University of Sofia, 8800 Sliven, Bulgaria
- Email: gstamov@abv.bg
- Received by editor(s): March 26, 2012
- Published electronically: November 4, 2013
- Communicated by: Yingfei Yi
- © Copyright 2013 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 142 (2014), 591-601
- MSC (2010): Primary 34K20; Secondary 34K25, 34K60
- DOI: https://doi.org/10.1090/S0002-9939-2013-12197-0
- MathSciNet review: 3134000