  We study time continuous branching processes with exponentially distributed
lifetimes, with two types of cells that proliferate according to binary
fission. A range of possible system dynamics are considered, each of which is
characterized by the mutation rate of the original cells and the survival
probability of the altered cells' progeny. For each system, we derive a
closed-form expression for the joint probability generating function of cell
counts, and perform asymptotic analysis on the behaviors of the cell population
with particular focus on probability of extinction. Part of our results
confirms known properties of branching processes using a different approach
while other are original. While the model is best suited for modeling the fate
of differentiating stem cells, we discuss other scenarios in which these system
dynamics may be applicable in real life. We also discuss the history of the
subject.
