Absolute and delay-dependent stability ofequations with a distributed delay

Elena Braverman; Sergey Zhukovskiy

doi:10.3934/dcds.2012.32.2041

Article Contents

2012, Volume 32, Issue 6: 2041-2061. Doi: 10.3934/dcds.2012.32.2041

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Absolute and delay-dependent stability of equations with a distributed delay

Elena Braverman^1, and
Sergey Zhukovskiy^2,

1.
Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, AB, T2N 1N4
2.
Department of Mathematics, Moscow PF University, Miklukho-Maklaya str. 6, Moscow 117198

Received: June 30, 2010

Revised: November 30, 2011

Published: May 2012

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Abstract

We study delay-independent stability in nonlinear models with a distributed delay which have a positive equilibrium. Models with a unique positive equilibrium frequently occur in population dynamics and other applications. In particular, we construct a relevant difference equation such that its stability implies stability of the equation with a distributed delay and a finite memory. This result is, generally speaking, incorrect for systems with infinite memory. If the relevant difference equation is unstable, we describe the general delay-independent lower and upper solution bounds and also demonstrate that the equation with a distributed delay is stable for small enough delays.

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Mathematics Subject Classification: Primary: 34K20; Secondary: 92D25, 34K60, 34K23.

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