×

A new approach to the global asymptotic stability problem in a delay Lotka-Volterra differential equation. (English) Zbl 1042.34571

Summary: In this paper, some new global attractivity results are given for scalar Lotka-Volterra differential equation with delays. Some examples and related open problems are also raised up at the end of the paper.

MSC:

34K20 Stability theory of functional-differential equations
34K60 Qualitative investigation and simulation of models involving functional-differential equations
Full Text: DOI

References:

[1] Gopalsamy, K., Stability and Oscillations in Delay Differential Equations of Population Dynamics (1992), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0752.34039
[2] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press Boston, MA · Zbl 0777.34002
[3] Kuang, Y., Global stability in delayed nonautonomous Lotka-Volterra type systems without saturated equilibria, Differential Integral Equations, 9, 3, 557-567 (1996) · Zbl 0843.34077
[4] Bereketoglu, H.; Györi, I., Global asymptotic stability in a nonautonomous Lotka-Volterra type system with infinite delay, J. Math. Anal. Appl., 210, 279-291 (1997) · Zbl 0880.34072
[5] He, X. Z., Global stability in nonautonomous Lotka-Volterra systems of “pure-delay type”, Diff. Int. Eqns., 11, 293-310 (1998) · Zbl 1022.34068
[6] Smith, H. L., Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Systems (1995), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0821.34003
[7] Györi, I., Asymptotic stability and oscillations of linear non-autonomous delay differential equations, (Proceedings of International Conference. Proceedings of International Conference, Columbus, OH, March 21-25 (1989), University Press: University Press Athens), 389-397 · Zbl 0717.34082
[8] Györi, I., Interaction between oscillation and global asymptotic stability in delay differential equations, Differential and Integral Equations, 3, 181-200 (1990) · Zbl 0726.34054
[9] Györi, I., Global attractivity in nonoscillating perturbed linear delay differential equations, (Proceedings of the International Symposium on Functional Differential Equations and Related Topics. Proceedings of the International Symposium on Functional Differential Equations and Related Topics, August 30-September 2, 1990, Kyoto, Japan (1991), World Scientific: World Scientific Singapore), 95-101 · Zbl 0796.34066
[10] Györi, I.; Pituk, M., Comparison theorems and asymptotic equilibrium for delay differential and difference equations, Dynamics Systems and Applications, 5, 277-302 (1996) · Zbl 0859.34053
[11] Bellman, R.; Cooke, K. L., Differential Difference Equations (1963), Academic Press: Academic Press New York · Zbl 0118.08201
[12] Hale, J. K., Theory of Functional Differential Equations (1977), Springer-Verlag · Zbl 0425.34048
[13] Györi, I.; Ladas, G., Oscillation Theory of Delay Differential Equations: With Applications (1991), Oxford University Press: Oxford University Press Oxford · Zbl 0780.34048
[14] Lenhart, S. M.; Travis, C. C., Global stability of a biological model with time delay, (Proc. Amer. Math. Soc., 96 (1986)), 75-78 · Zbl 0602.34044
[15] Jianhua, S.; Zhicheng, W., Global attractivity in a nonautonomous delay-logistic equation, Tamkang Journal of Mathematics, 26, 159-164 (1995) · Zbl 0838.34089
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.