  An individual-based model of stochastic branching is proposed and studied, in
which point particles drift in $\bar{\mathds{R}}_{+}:=[0,+\infty)$ towards the
origin (edge) with unit speed, where each of them splits into two particles
that instantly appear in $\bar{\mathds{R}}_{+}$ at random positions. During
their drift the particles are subject to a random disappearance (death). The
model is intended to capture the main features of the proliferation of tumor
cells, in which trait $x\in \bar{\mathds{R}}_{+}$ of a given cell is time to
its division and the death is caused by therapeutic factors. The main result of
the paper is proving the existence of an honest evolution of this kind and
finding a condition that involves the death rate and cell cycle distribution
parameters, under which the mean size of the population remains bounded in
time.
