  We propose a model of run-and-tumble particles (RTPs) on a line with a
fertile site at the origin. After going through the fertile site, a
run-and-tumble particle gives rise to new particles until it flips direction.
The process of creation of new particles is modelled by a fertility function
(of the distance to the fertile site), multiplied by a fertility rate. If the
initial conditions correspond to a single RTP with even probability density,
the system is parity-invariant. The equations of motion can be solved in the
Laplace domain, in terms of the density of right-movers at the origin. At large
time, this density is shown to grow exponentially, at a rate that depends only
on the fertility function and fertility rate. Moreover, the total density of
RTPs (divided by the density of right-movers at the origin), reaches a
stationary state that does not depend on the initial conditions, and presents a
local minimum at the fertile site.
