$\rho = \rho_A + \rho_B$, $\tilde{\rho}_A = 1-\rho_A$ and $\tilde{\rho}_B = 1-\rho_B$. If one species is absent (for example, if $p_B = p_{AB} = 0$) this system reduces to the MFT for the basic contact process, $\dot{p}_A = \lambda p_A (1-p_A) - p_A$, with a critical point at $\lambda = 1$. Under the effect of symbiosis we seek a symmetric stationary solution, $p_A = p_B = p$, leading, for $D=0$, to

$$\overline{p} = \frac{\mu}{2 \lambda (1-\mu)} \left[ 2(1-\mu) - \lambda
+ \sqrt{\lambda^2 - 4\mu (1-\mu)} \right].
\label{pMFT}$$

and $$\overline{p}_{AB} = \frac{\lambda p^2}{\mu - \lambda p}
\label{pabmft}$$

For $\mu \geq 1/2$, $p$ grows continuously from zero at $\lambda=1$, marking the latter value as the critical point. The activity grows linearly, $p \simeq [\mu/(2\mu -1)](\lambda-1)$, in this regime. For $\mu < 1/2$, however, the expression is already positive for $\lambda = \sqrt{4 \mu(1-\mu)} < 1$, and there is a [*discontinuous*]{} transition at this point.

In the limit $D \to \infty$, we expect $p_{AB} = \rho_A \rho_B$, as is required by the condition that, in this limit, a time-independent solution requires that the coefficient of $D$ in Eq. \[pab\] be zero.

Cluster approximations and quasistationary analysis
===================================================

As noted above, the discontinuous phase 