×

zbMATH — the first resource for mathematics

Harmless delays for uniform persistence. (English) Zbl 0731.34085
The authors consider a predator-prey system of Lotka-Volterra type with a finite number of discrete delays. They give sufficient conditions that the system either be uniformly persistent or not persistent. These conditions are the same as the corresponding conditions for the system with all delays set equal to zero and are independent of the magnitude of the delays. Hence uniform persistence of a system is not affected by the time delays. The results are obtained by construction of suitable Lyapunov functionals.
If the condition for persistence is satisfied, the system possesses an equilibrium. In this case, the average of the solution approaches the equilibrium.
Finally, the authors give an example of a system in which the stability of the equilibrium changes as the delay varies, but the persistence of the system is unaffected.
Reviewer: A.Hausrath (Boise)

MSC:
34K20 Stability theory of functional-differential equations
92D25 Population dynamics (general)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] May, R.M, Time delay versus stability in population models with two or three trophic levels, Ecology, 54, No. 2, 315-325, (1973)
[2] Gopalsamy, K, Harmless delays in model ecosystems, Bull. math. biol., 45, 295-309, (1983) · Zbl 0514.34060
[3] Freedman, H.I; Gopalsamy, K, Nonoccurence of stability switching in systems with discrete delays, Canad. math. bull., 31, No. 1, 52-58, (1988) · Zbl 0607.34062
[4] Freedman, H.I; Rao, V.S.H, Stability criteria for a system involving two time delays, SIAM. J. appl. math., 46, 552-560, (1986) · Zbl 0624.34066
[5] Gopalsamy, K; Aggarwala, B.D, Limit cycles in two species competition with time delays, J. austral. soc. ser. B, 22, 148-160, (1980) · Zbl 0458.92014
[6] Blythe, S.P; Nisbet, R.M; Gurney, W.S.C; Macdonald, N, Stability switches in distributed delay models, J. math. anal. appl., 109, 388-396, (1985) · Zbl 0589.92018
[7] Cooke, K.L; Grossman, Z, Discrete delay, distributed delay and stability switches, J. math. anal. appl., 86, 592-627, (1982) · Zbl 0492.34064
[8] Zhien, M, Stability of predation models with time delays, Appl. anal., 22, 169-192, (1986) · Zbl 0592.92020
[9] Lensu, C, Mathematical models and methods in ecology, (1988), Science Press Beijing, [in Chinese]
[10] Serifert, George, On a delay-differential equation for single species populations, Nonlinear anal., 11, No. 9, 1051-1059, (1987) · Zbl 0629.92019
[11] Hale, J.K, Theory of functional differential equations, (1977), Springer-Verlag New York · Zbl 0425.34048
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.