STABILITY ANALYSIS OF AN EQUATION WITH TWO DELAYS AND APPLICATION TO THE PRODUCTION OF PLATELETS

摘要

We analyze the stability of a differential equation with two delays originating from a model for a population divided into two subpopulations, immature and mature, and we apply this analysis to a model for platelet production. The dynamics of mature individuals is described by the following nonlinear differential equation with two delays: x'(t) =-γx(t)+g(x(t-τ_1))-g(x(t-τ_1-τ_2))e~(-γτ_2) . The method of D-decomposition is used to compute the stability regions for a given equilibrium. The centre manifold theory is used to investigate the steady-state bifurcation and the Hopf bifurcation. Similarly, analysis of the centre manifold associated with a double bifurcation is used to identify a set of parameters such that the solution is a torus in the pseudophase space. Finally, the results of the local stability analysis are used to study the impact of an increase of the death rate or of a decrease of the survival time 2 of platelets on the onset of oscillations. We show that the stability is lost through a small decrease of survival time (from 8.4 to 7 days), or through an important increase of the death rate (from 0.05 to 0.625 days-1).