  Motivated by experimental and theoretical work on autonomous oscillations in
yeast, we analyze ordinary differential equations models of large populations
of cells with cell-cycle dependent feedback. We assume a particular type of
feedback that we call Responsive/Signaling (RS), but do not specify a
functional form of the feedback. We study the dynamics and emergent behaviour
of solutions, particularly temporal clustering and stability of clustered
solutions. We establish the existence of certain periodic clustered solutions
as well as "uniform" solutions and add to the evidence that cell-cycle
dependent feedback robustly leads to cell-cycle clustering. We highlight the
fundamental differences in dynamics between systems with negative and positive
feedback. For positive feedback systems the most important mechanism seems to
be the stability of individual isolated clusters. On the other hand we find
that in negative feedback systems, clusters must interact with each other to
reinforce coherence. We conclude from various details of the mathematical
analysis that negative feedback is most consistent with observations in yeast
experiments.
