We illustrate the appearance of oscillating solutions in delay differential equations
modeling hematopoietic stem cell dynamics. We focus on autonomous oscillations, arising as
consequences of a destabilization of the system, for instance through a Hopf bifurcation.
Models of hematopoietic stem cell dynamics are considered for their abilities to describe
periodic hematological diseases, such as chronic myelogenous leukemia and cyclical
neutropenia. After a review of delay models exhibiting oscillations, we focus on three
examples, 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 oscillating
solutions, and we characterize these solutions in terms of periods and amplitudes.