Periodic Oscillations of Blood Cell Populations in Chronic Myelogenous Leukemia

We develop some techniques to prove analytically the existence and stability of long period oscillations of stem cell populations in the case of periodic chronic myelogenous leukemia. Such a periodic oscillation $p_\infty $ can be analytically constructed when the hill coefficient involved in the nonlinear feedback is infinite, and we show it is possible to obtain a contractive returning map (for the semiflow defined by the modeling functional differential equation) in a closed and convex cone containing $p_\infty $ when the hill coefficient is large, and the fixed point of such a contractive map gives the long period oscillation previously observed both numerically and experimentally.