×

zbMATH — the first resource for mathematics

Extinction in a two dimensional Lotka-Volterra system with infinite delay. (English) Zbl 1122.34058
A two-dimensional competitive nonautonomous Lotka-Volterra system is considered with infinite delay for the second variable in the first equation and vice-versa in the second one. A result of S. Ahmad [Proc. Am. Math. Soc. 117, No. 1, 119–204 (1993; Zbl 0843.34033)] is extended to the present case. The corresponding inequality implies the elimination of the second competitor and the survival of the first one.

MSC:
34K25 Asymptotic theory of functional-differential equations
92D25 Population dynamics (general)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ahmad, S., On the nonautonomous volterra – lotka competition equations, Proc. am. math. soc., 117, 119-204, (1993) · Zbl 0848.34033
[2] Ahmad, S.; Rao, M.R.M., Asymptotically periodic solutions of N-competing species problem with time delays, J. math. anal. appl., 186, 559-571, (1994) · Zbl 0818.45004
[3] Ahmad, S.; Rao, M.R.M., Stability criteria for N-competing species problem with time delays, Nonlinear world, 245-253, (1994) · Zbl 0798.92022
[4] Burton, T.A., Stability and periodic solutions od ordinary and functional differential equations, (1985), Academic Press San Diego · Zbl 0635.34001
[5] Driver, R., Existence and stability of solutions of delay differential equations, Arch. rational mech. anal., 10, 401-426, (1962) · Zbl 0105.30401
[6] Gopalsamy, K., Time lags and global stability in two species competition, Bull. math. biol., 42, 729-737, (1980) · Zbl 0453.92014
[7] Gopalsamy, K., Global asymptotic stability in Volterra’s population systems, J. math. biol., 19, 157-168, (1984) · Zbl 0535.92020
[8] Gopalsamy, K., Global asymptotic stability in a class of volterra – stieltjes integrodifferential systems, Int. J. systems sci., 18, 1733-1737, (1987) · Zbl 0634.45007
[9] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Publishers Dordrecht · Zbl 0752.34039
[10] Hale, J.K., Theory of functional differential equations, (1977), Springer New York
[11] Hirsch, W.; Hanisch, H.; Gabriel, J., Differential equation models of some parasitic infection-methods for the study of asymptotic behavior, Comm. pure appl. math., 38, 733-753, (1985) · Zbl 0637.92008
[12] B. Lisena, Competitive exclusion in a periodic Lotka-Volterra system, J. Appl. Math. Comput., to appear. · Zbl 1100.92070
[13] B. Lisena, Asymptotic behaviour in periodic three species predator-prey system, Annali di Matematica Pura ed Applicata., to appear. · Zbl 1232.34082
[14] Tineo, A., Asymptotic behaviour of positive solutions of the nonautonomous lotka – volterra competition equations, Differential and integral equations, 6, 419-457, (1993) · Zbl 0774.34037
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.