Stability and Hopf Bifurcation Analysis for Functional Differential Equation with Distributed Delay

Abstract

In some applications of delay differential equations in population dynamics, the need for incorporation of distributed delay is often the result of the existence of some processes or structures. The main objective of this paper is to provide some information on the stability and Hopf bifurcation analysis in a general functional differential equation with distributed delay. The local stability parameter regions related to time delay are given and compared for a general distribution delay function and three frequently used distributed delays including Dirac, uniform, and Gamma distributions. With the loss of stability at the boundary of these regions, we discuss the Hopf bifurcation using a normal-form method; there the computation of the coefficients is given explicitly in the form of the corresponding characteristic equation. Using the theoretical results, we study two examples of white blood cell models and address the effect of the distributed time delay on the physiological oscillations. Numerical simu...