  In this paper, we introduce a one-dimensional model of particles performing
independent random walks, where only pairs of particles can produce offspring
("cooperative branching"), and particles that land on an occupied site merge
with the particle present on that site ("coalescence"). We show that the system
undergoes a phase transition as the branching rate is increased. For small
branching rates, the upper invariant law is trivial, and the process started
with finitely many particles a.s. ends up with a single particle. Both
statements are not true for high branching rates. An interesting feature of the
process is that the spectral gap is zero even for low branching rates. Indeed,
if the branching rate is small enough, then we show that for the process
started in the fully occupied state, the particle density decays as one over
the square root of time, and the same is true for the decay of the probability
that the process still has more than one particle at a later time if it started
with two particles.
