  We study a continuous time Mutually Catalytic Branching model on the
$\mathbb{Z}^{d}$. The model describes the behavior of two different populations
of particles, performing random walk on the lattice in the presence of
branching, that is, each particle dies at a certain rate and is replaced by a
random number of offsprings. The branching rate of a particle in one population
is proportional to the number of particles of another population at the same
site. We study the long time behavior for this model, in particular,
coexistence and non-coexistence of two populations in the long run. Finally, we
construct a sequence of renormalized processes and use duality techniques to
investigate its limiting behavior.
