Periodic Solutions in a Differential Delay Equation Modeling Megakaryopoiesis

We consider a scalar nonlinear differential delay equation which was recently proposed as a mathematical model for platelet production (megakaryopoiesis). The equation has a unique positive equilibrium about which solutions tend to oscillate. We show that periodic oscillations in the model always exist when the equilibrium is linearly unstable. Several methods of proof are proposed. They include an adapted version of established ejective fixed point techniques, and application of a more recent theorem for nonlinear semiflows. We indicate how an analogous result can be obtained for a different class of equations frequently used in applications.