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

Optimal linear stability condition for scalar differential equations with distributed delay