zbMATH — the first resource for mathematics

Convergence of solutions for Volterra-Lotka prey-predator systems with time delays. (English) Zbl 1163.34303
Summary: This work is concerned with \(N\)-species prey-predator systems with time delays. The aim of this work is to obtain a sufficient condition for asymptotic behavior of the time-dependent solution and the existence of a positive steady-state solution. The result of global asymptotic stability implies that all of the model systems coexist; the trivial and all kinds of semitrivial solutions are unstable.

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
92D25 Population dynamics (general)
Full Text: DOI
[1] Li, L.; Cao, G.; Liu, Y., The relationships between equilibria and positive solutions of certain nonlinear elliptic systems, J. math. anal. appl., 209, 154-179, (1997) · Zbl 0884.35032
[2] Li, L., On positive solutions of general nonlinear elliptic symbiotic interacting systems, Appl. anal., 40, 281-295, (1991) · Zbl 0757.35023
[3] Yamada, Y., Stability of steady-states for prey – predator diffusion equations with homogeneous Dirichlet conditions, SIAM J. math. anal., 21, 327-345, (1990) · Zbl 0702.35123
[4] Zhou, L.; Pao, C.V., Asymptotic behavior for a competition – diffusion system in population dynamics, Nonlinear anal., 6, 1163-1184, (1982) · Zbl 0522.92017
[5] Feng, W., Permanence effect in a three species food chain model, Appl. anal., 54, 195-209, (1994) · Zbl 0834.92023
[6] Lakos, N., Existence of steady-state solutions for a one-predator – two-prey system, SIAM J. math. anal., 21, 647-659, (1990) · Zbl 0705.92019
[7] Pao, C.V., Nonlinear parabolic and elliptic equations, (1992), Academic Press New York · Zbl 0780.35044
[8] Pao, C.V., Quasisolutions and global attractor of reaction – diffusion systems, Nonlinear anal., 26, 1889-1903, (1996) · Zbl 0853.35056
[9] Ruan, S.G.; Zhao, X.Q., Persistence and extinction in two species reaction diffusion systems with delays, J. differential equations, 156, 71-92, (1999) · Zbl 0936.35092
[10] Ruan, W.; Feng, W., On the fixed point index and multiple steady-state solutions of reaction diffusion system, Differ. integral equ., 2, 371-391, (1995) · Zbl 0815.35017
[11] Pao, C.V., Global asymptotic stability of lotka – volterra 3-species reaction – diffusion systems with time delays, J. math. anal. appl., 281, 186-204, (2003) · Zbl 1031.35071
[12] Pao, C.V., Convergence of solutions of reaction – diffusion systems with time delays, Nonlinear anal., 48, 349-362, (2002) · Zbl 0992.35105
[13] Pao, C.V., Global asymptotic stability of lotka – volterra competition systems with diffusion and time delays, Nonlinear anal. RWA, 5, 91-104, (2004) · Zbl 1066.92054
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.