Degenerate Lyapunov functional and a prey-predator model with infinite delays


Xue-Zhong He


Research Report 97-14
Date: 7 May 1997


The effect of delays on the stability is one of the most fundamental problems in delay differential equations (DDEs). It is commonly believed that small delays are negligible for a stable DDEs, however, to have an affirmative answer is still difficult in general. In this paper, the asymptotic behavior of the classical Lotka-Volterra prey-predator integrodifferential equations with infinite delays is concerned by an approach of perturbation and degenerate Lyapunov functionals. We first show that all the positive solutions of the system is bounded above eventually. Then, by treating the system as a perturbation of the corresponding ODE system and constructing suitable degenerate Lyapunov functionals, we are able to obtain some sufficient conditions on the global attractivity of the positive equilibrium. As a corollary, we show that small delays do not change the stability of the system.

Key phrases

global attractivity. degenerate Lyapunov functionals. prey-predator model. infinite delays.

AMS Subject Classification (1991)

Primary: 34D05, 34E10, 34K20, 45J05, 92A15, 92D25
Secondary: 55R10, 53C30


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