  A biochemical oscillator model, describing developmental stage of
myxobacteria, is analyzed mathematically. Observations from numerical
simulations show that in a certain range of parameters, the corresponding
system of ordinary differential equations displays stable and robust
oscillations. In this work, we use geometric singular perturbation theory and
blow-up method to prove the existence of a strongly attracting limit cycle.
This cycle corresponds to a relaxation oscillation of an auxiliary system,
whose singular perturbation nature originates from the small Michaelis-Menten
constants of the biochemical model. In addition, we give a detailed description
of the structure of the limit cycle, and the timescales along it.
