Approximating the Stability Region for a Differential Equation with a Distributed Delay

S. A. Campbell; R. Jessop

Mathematical Modelling of Natural Phenomena (2009)

  • Volume: 4, Issue: 2, page 1-27
  • ISSN: 0973-5348

Abstract

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

How to cite

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Campbell, S. A., and Jessop, R.. "Approximating the Stability Region for a Differential Equation with a Distributed Delay." Mathematical Modelling of Natural Phenomena 4.2 (2009): 1-27. <http://eudml.org/doc/222208>.

@article{Campbell2009,
abstract = { We discuss how distributed delays arise in biological models and review the literature on such models. We indicate why it is important to keep the distributions in a model as general as possible. We then demonstrate, through the analysis of a particular example, what kind of information can be gained with only minimal information about the exact distribution of delays. In particular we show that a distribution independent stability region may be obtained in a similar way that delay independent results are obtained for systems with discrete delays. Further, we show how approximations to the boundary of the stability region of an equilibrium point may be obtained with knowledge of one, two or three moments of the distribution. We compare the approximations with the true boundary for the case of uniform and gamma distributions and show that the approximations improve as more moments are used. },
author = {Campbell, S. A., Jessop, R.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {delay differential equations; distributed delay; linear stability; delay independent stability; delay independent stability; approximation of stability region},
language = {eng},
month = {3},
number = {2},
pages = {1-27},
publisher = {EDP Sciences},
title = {Approximating the Stability Region for a Differential Equation with a Distributed Delay},
url = {http://eudml.org/doc/222208},
volume = {4},
year = {2009},
}

TY - JOUR
AU - Campbell, S. A.
AU - Jessop, R.
TI - Approximating the Stability Region for a Differential Equation with a Distributed Delay
JO - Mathematical Modelling of Natural Phenomena
DA - 2009/3//
PB - EDP Sciences
VL - 4
IS - 2
SP - 1
EP - 27
AB - We discuss how distributed delays arise in biological models and review the literature on such models. We indicate why it is important to keep the distributions in a model as general as possible. We then demonstrate, through the analysis of a particular example, what kind of information can be gained with only minimal information about the exact distribution of delays. In particular we show that a distribution independent stability region may be obtained in a similar way that delay independent results are obtained for systems with discrete delays. Further, we show how approximations to the boundary of the stability region of an equilibrium point may be obtained with knowledge of one, two or three moments of the distribution. We compare the approximations with the true boundary for the case of uniform and gamma distributions and show that the approximations improve as more moments are used.
LA - eng
KW - delay differential equations; distributed delay; linear stability; delay independent stability; delay independent stability; approximation of stability region
UR - http://eudml.org/doc/222208
ER -

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