We illustrate the appearance of oscillating solutions in delay differential equationsmodeling hematopoietic stem cell dynamics. We focus on autonomous oscillations, arising asconsequences of a destabilization of the system, for instance through a Hopf bifurcation.Models of hematopoietic stem cell dynamics are considered for their abilities to describeperiodic hematological diseases, such as chronic myelogenous leukemia and cyclicalneutropenia. After a review of delay models exhibiting oscillations, we focus on threeexamples, describing different delays: a discrete delay, a continuous distributed delay,and a state-dependent delay. In each case, we show how the system can have oscillatingsolutions, and we characterize these solutions in terms of periods and amplitudes.
