# zbMATH — the first resource for mathematics

Global asymptotic stability in $$n$$-species nonautonomous Lotka-Volterra competitive systems with infinite delays. (English) Zbl 1029.34060
The authors consider $$n$$-species nonautonomous Lotka-Volterra-type competitive systems with infinite delay. By means of a Lyapunov functional, they prove the global asymptotic stability of any positive solution to the system.
Moreover, conditions are given such that the positive solution to the delayed system is globally asymptotically stable whenever the positive solution to the nondelayed system is globally asymptotically stable.
An example is given to illustrate the feasibility of the main result.

##### MSC:
 34K20 Stability theory of functional-differential equations 92D25 Population dynamics (general)
Full Text:
##### References:
 [1] Ahmad, S.; Mohana Rao, M.R., Asymptotically periodic solutions of n-competing species problem with time delay, J. math. anal. appl., 186, 557-571, (1994) · Zbl 0818.45004 [2] Freedman, H.I.; Ruan, S., Uniform persistence in functional differential equations, J. differential equations, 115, 173-192, (1995) · Zbl 0814.34064 [3] Kuang, Y.; Smith, H.L., Global stability for infinite delay lotka – volterra type systems, J. differential equations, 103, 221-246, (1993) · Zbl 0786.34077 [4] Kuang, Y.; Tang, B., Uniform persistence in nonautonomous delay differential Kolmogorov type population models, Rocky mountain J. math., 24, 165-186, (1994) · Zbl 0823.92021 [5] Kuang, Y., Global stability in delay differential systems without dominating instantaneous negative feedbacks, J. differential equations, 119, 503-532, (1995) · Zbl 0828.34066 [6] Kuang, Y., Global stability in delayed nonautonomous lotka – volterra type systems without saturated equilibria, Differential integral equations, 9, 557-567, (1996) · Zbl 0843.34077 [7] Tang, B.; Kuang, Y., Permanence in Kolmogorov type systems of nonautonomous functional differential equations, J. math. anal. appl., 197, 427-447, (1996) · Zbl 0951.34051 [8] Wang, W.; Chen, L.; Lu, Z., Global stability of a competition model with periodic coefficients and time delays, Canad. appl. math. quarterly, 3, 365-378, (1995) · Zbl 0845.92020 [9] Wang, L.; Zhang, Y., Global stability of volterra – lotka systems with delay, Differential equations dynam. sys., 3, 205-216, (1995) · Zbl 0880.34074 [10] Bereketoglu, H.; Gyori, I., Global asymptotic stability in a nonautonomous lotka – volterra type systems with infinite delay, J. math. anal. appl., 210, 279-291, (1997) · Zbl 0880.34072 [11] Teng, Z.; Yu, Y., Some new results of nonautonomous lotka – volterra competitive systems with delays, J. math. anal. appl., 241, 254-275, (2000) · Zbl 0947.34066 [12] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Publishers Dordrecht · Zbl 0752.34039 [13] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press Boston, MA · Zbl 0777.34002 [14] Hale, J., Theory of functional differential equations, (1977), Springer Heidelberg
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.