  This theoretical study concerns a pH oscillator based on the urea-urease
reaction confined to giant lipid vesicles. Under suitable conditions,
differential transport of urea and hydrogen ion across the unilamellar vesicle
membrane periodically resets the pH clock that switches the system from acid to
basic, resulting in self-sustained oscillations. We analyse the structure of
the phase flow and of the limit cycle, which controls the dynamics for giant
vesicles and dominates the pronouncedly stochastic oscillations in small
vesicles of submicrometer size. To this end, we derive reduced models, which
are amenable to analytic treatments that are complemented by numerical
solutions, and obtain the period and amplitude of the oscillations as well as
the parameter domain, where oscillatory behavior persists. We show that the
accuracy of these predictions is highly sensitive to the employed reduction
scheme. In particular, we suggest an accurate two-variable model and show its
equivalence to a three-variable model that admits an interpretation in terms of
a chemical reaction network. The faithful modeling of a single pH oscillator
appears crucial for rationalizing experiments and understanding communication
of vesicles and synchronization of rhythms.
