continuous in this case, the overall density is very low near the critical point. In the presence of diffusion, pairs tend to be destroyed; the resulting isolated individuals then rapidly die. Thus diffusion renders low-density active states inviable. Under moderate diffusion, a finite density is required to maintain a significant concentration of doubly occupied sites, and thereby maintain activity. Hence the population density jumps from zero to a finite value at the transition. For small $\mu$ we observe a discontinuous phase transition even for small values of the diffusion rate, as shown in Figs. \[mu01d.1\] and \[rqd01\].

![[]{data-label="crsurf"}](crsurf.eps){width="0.95\hsize"}

Although we have verified that the phase transition is discontinuous for small $\mu$ and moderate diffusion rates $D$, increasing $D$ further, the transition becomes continuous again. In the limit $D \to \infty$, we expect mean field-like behavior, with the effects of diffusion suppressing the clustering which permits symbiosis. In this limit, the one-site MFT predicts a [*continuous*]{} phase transition, with $\lambda_c = 1$, for any value of $\mu$. Reversion to a continuous transition under rapid diffusion ($D=100$, $\mu=0.25$) is evident in Fig. \[mu25d100\]: the QS probability distributions are again unimodal. At criticality, fewer than $4\%$ of the individuals are located at doubly 