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

Optimal linear stability condition for scalar differential equations with distributed delay

Pages: 1855 - 1876, Volume 20, Issue 7, September 2015      doi:10.3934/dcdsb.2015.20.1855

 
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Samuel Bernard - Université de Lyon; CNRS UMR 5208, Université Lyon 1; Institut Camille Jordan, INRIA Team Dracula, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France (email)
Fabien Crauste - Université de Lyon; CNRS UMR 5208, Université Lyon 1; Institut Camille Jordan, INRIA Team Dracula, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France (email)

Abstract: Linear scalar differential equations with distributed delays appear in the study of the local stability of nonlinear differential equations with feedback, which are common in biology and physics. Negative feedback loops tend to promote oscillations around steady states, and their stability depends on the particular shape of the delay distribution. Since in applications the mean delay is often the only reliable information available about the distribution, it is desirable to find conditions for stability that are independent from the shape of the distribution. We show here that for a given mean delay, the linear equation with distributed delay is asymptotically stable if the associated differential equation with a discrete delay is asymptotically stable. We illustrate this criterion on a compartment model of hematopoietic cell dynamics to obtain sufficient conditions for stability.

Keywords:  Delay differential equations, delay distribution, negative feedback loop, hematopoiesis.
Mathematics Subject Classification:  Primary: 34K06, 34K20; Secondary: 92C30.

Received: April 2014;      Revised: February 2015;      Available Online: July 2015.

 References