  A competition process on $\mathbb{Z}^d$ is considered, where two species
compete to color the sites. The entities are driven by branching random walks.
Specifically red (blue) particles reproduce in discrete time and place
offspring according to a given reproduction law, which may be different for the
two types. When a red (blue) particle is placed at a site that has not been
occupied by any particle before, the site is colored red (blue) and keeps this
color forever. The types interact in that, when a particle is placed at a site
of opposite color, the particle adopts the color of the site with probability
$p\in[0,1]$. Can a given type color infinitely many sites? Can both types color
infinitely many sites simultaneously? Partial answers are given to these
questions and many open problems are formulated.
