  We study the effects of hard core (HC) interactions between different species
of particles on two-species branching annihilating random walks with one
offspring(BAW$_2$(1)). The single-species model belongs to the directed
percolation (DP) universality class. In the BAW$_2$(1) model, a particle
creates one particle of the same species in its neighborhood with the
probability $\sigma (1-p)$ and of the different species with $(1-\sigma)(1-p)$,
where $p$ is the hopping probability. Without HC interactions, this model
always exhibits the DP-type absorbing transition for all $\sigma$. Even with HC
interactions, the nature of the phase transitions does not change except near
$\sigma=0$, where the HC interaction destabilizes and completely wipes away the
absorbing phase. The model is always active except at the annihilation fixed
point of zero branching rate ($p=1$). Critical behavior near the annihilation
fixed point is characterized by exponents $\beta = \nu_\perp =1/2$ and
$\nu_{||} = 1$.
