  In this paper we study, at different levels of generality, certain systems of
delay differential equations (DDE). One focus and motivation is a system with
state-dependent delay (SD-DDE) that has been formulated to describe the
maturation of stem cells. We refer to this system as the cell SD-DDE. In the
cell SD-DDE, the delay is implicitly defined by a threshold condition. The
latter is specified by the time at which the (also implicitly defined) solution
of an external nonlinear ordinary differential equation (ODE), which is
parametrised by a component of the SD-DDE, meets a given threshold value. We
focus on the dynamical properties global asymptotic stability (GAS) of the zero
equilibrium, persistence and dissipativity/ultimate boundedness.
