Our first goal in this work is to give a proof of exchange of stability from the trivialbranch to the bifurcated one. This proof is based on the two following steps:i) reduction of the equation to a two-dimensional system via the variation of constantformula end the center manifold theorem.ii) Estimation of the distance between solutions of the original equation and the bifurcatedperiodic solutions.We obtain an estimate of the stability region.The second goal is to study the dynamics of Haematopoietic Stem Cells (HSC) Modelwith one delay.The model, was initially introduced by Mackey (1978). There are two possible stationarystates. One of them is trivial and unstable, the second is nontrivial, depending onthe delay \tau.We prove the existence of a critical value ¿0 of the delay \tau in which the exchange ofstability of nontrivial stationary state may occur.We introduce also an approachable model depending on this critical value of the delay,such that the nontrivial stationary state do not depend on the delay which is the sameone of Mackey model at \tau =\tau_{0}.By a similar study of the approachable model as in Mackey model, we obtain the existenceof the bifurcated periodic solution branch around the nontrivial stationary state.In the end, we give an explicit algorithm for calculating the elements of bifurcation.
