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Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

Sufficient conditions for stability of linear differential equations with distributed delay

Pages: 233 - 256, Volume 1, Issue 2, May 2001      doi:10.3934/dcdsb.2001.1.233

 
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Samuel Bernard - Département de Mathématiques et de Statistique and Centre de recherches mathématiques, Université de Montréal, Montréal Québec H3C 3J7, Canada (email)
Jacques Bélair - Département de Mathématiques et de Statistique Centre de recherches mathématiques and Institut de Génie Biomédical, Université de Montréal, Montréal Québec H3C 3J7, Canada (email)
Michael C Mackey - Departments of Physiology, Physics & Mathematics and Centre for Nonlinear Dynamics, McGill University, 3655 Drummond, Montréal, Québec H3G 1Y6, Canada (email)

Abstract: We develop conditions for the stability of the constant (steady state) solutions oflinear delay differential equations with distributed delay when only information about the moments of the density of delays is available. We use Laplace transforms to investigate the properties of different distributions of delay. We give a method to parametrically determine the boundary of the region of stability, and sufficient conditions for stability based on the expectation of the distribution of the delay. We also obtain a result based on the skewness of the distribution. These results are illustrated on a recent model of peripheral neutrophil regulatory system which include a distribution of delays. The goal of this paper is to give a simple criterion for the stability when little is known about the distribution of the delay.

Keywords:  Differential equations, distributed delay, stability, cyclical neutropenia.
Mathematics Subject Classification:  34K20, 34K06, 92C37.

Received: November 2000;      Revised: January 2001;      Available Online: February 2001.