transition predicted by one- and two-site MFT is impossible in one dimension. Simulations in both one and two dimensions, covering a broad range of $\mu$ values, yield no evidence of a discontinuous transition. Here we attempt to develop more reliable theoretical descriptions, using cluster approximations and quasistationary (QS) solutions of small systems, for the symmetric case, $\lambda_A = \lambda_B = \lambda$. In the following analysis we set $D=0$, i.e., the non-diffusive limit of Eqs. 1 - 4.

It is often the case that MFT predictions improve, both qualitatively and quantitatively, as the cluster size used in the analysis is increased. We therefore investigate MFT approximations using clusters of up to six sites in one dimension, and clusters of four sites on the square lattice. Following the usual procedure [@mftzgb; @marro; @benav], we deduce a set of coupled, nonlinear differential equations for the cluster occupation probabilities, which are then integrated numerically to obtain the stationary solution. As shown in Fig. \[lm1d\], for the one-dimensional case, the prediction for the phase boundary in the $\lambda-\mu$ plane does improve as we increase the cluster size from $n=2$ to $n=6$. The $n=2$ approximation correctly predicts a continuous phase transition for $\mu \geq 0.75$, but 