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Degenerate Lyapunov functionals of a well-known prey–predator model with discrete delays

Published online by Cambridge University Press:  14 November 2011

Xue-Zhong He
Xue-Zhong He
Affiliation:
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
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Extract

It is commonly believed that, as far as stabilities are concerned, ‘small delays are negligible in some modelling processes’. However, to have an affirmative answerfor this common belief is still an open problem for many nonlinear equations. In this paper, the classical Lotka–Volterra prey–predator equation with discrete delays

is considered, and, by using degenerate Lyapunov functionals method, an affirmative answer to this open problem on both local and global stabilities of the prey–predator delay equations is given. It is shown that degenerate Lyapunov functional method is a powerful tool for studying the stability of such nonlinear delay systems. A detailed and explicit procedure of constructing such functionals is provided. Furthermore, some explicit estimates on the allowable sizes of the delays are obtained.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 4 , 1999 , pp. 755 - 771
Copyright
Copyright © Royal Society of Edinburgh 1999

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