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Approximating the Stability Region for a Differential Equationwith a Distributed Delay

Published online by Cambridge University Press:  26 March 2009

S. A. Campbell and
R. Jessop
S. A. Campbell*
Affiliation:
Department of Applied Mathematics, University of Waterloo, Waterloo, N2L 3G1, Canada
R. Jessop
Affiliation:
Department of Applied Mathematics, University of Waterloo, Waterloo, N2L 3G1, Canada
*
sacampbell@uwaterloo.ca
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Abstract

We discuss how distributed delays arise in biological models and review theliterature on such models. We indicate why it is important to keep thedistributions in a model as general as possible. We then demonstrate, throughthe analysis of a particular example, what kind of information can be gainedwith only minimal information about the exact distribution of delays.In particular we show that a distribution independent stability region maybe obtained in a similar way that delay independent results are obtained forsystems with discrete delays. Further, we show how approximations to theboundary of the stability region of an equilibrium point may be obtained withknowledge of one, two or three moments of the distribution. We compare theapproximations with the true boundary for the case of uniform and gammadistributions and show that the approximations improve as more moments are used.

Keywords

delay differential equationsdistributed delaylinear stabilitydelay independent stability

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