  We study here the dynamics (and stability) of Probabilistic Population
Protocols, via the differential equations approach. We provide a quite general
model and we show that it includes the model of Angluin et. al. in the case of
very large populations. For the general model we give a sufficient condition
for stability that can be checked in polynomial time. We also study two
interesting subcases: (a) protocols whose specifications (in our terms) are
configuration independent. We show that they are always stable and that their
eventual subpopulation percentages are actually a Markov Chain stationary
distribution. (b) protocols that have dynamics resembling virus spread. We show
that their dynamics are actually similar to the well-known Replicator Dynamics
of Evolutionary Games. We also provide a sufficient condition for stability in
this case.
