understanding of the phase diagram of the 2SCP, using exact quasistationary probability distributions for small systems, cluster approximations, and simulations. In two dimensions, we extend the model to include diffusion (nearest-neighbor hopping) of individuals. While we find no evidence of a discontinuous transition without diffusion, it becomes discontinuous for sufficiently small $\mu$ and large $D$.

The remainder of this paper is organized as follows. In Sec. II we review the definition of the model and the mean-field analysis, and in Sec. III present results of cluster approximations and quasistationary analysis. Then, in Sec. IV we study the diffusive process. Sec. V is devoted to discussion and conclusions.

Model
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To begin we review the definition of the two-species symbiotic contact process (2SCP) [@scp]. We denote the variables for occupation of a site $i$ by species A and B as $\sigma_i$ and $\eta_i$, respectively. The possible states $(\sigma_i, \eta_i)$ of a given site are $(0,0)$ (empty), $(1,0)$ (occupied by species A only), $(0,1)$ (species B only), and $(1,1)$ (occupied by both species). Birth of $A$ individuals, represented by the transitions $(0,0) \to (1,0)$ and $(0,1) \to (1,1)$, occur at rate $\lambda_A r_A$, with $r_A$ the fraction of nearest neighbor sites (NNs) 