that the moment ratio $m(\lambda;L) \equiv \langle \rho^2 \rangle/\langle \rho \rangle^2$ exhibits crossings, analogous to those of the Binder cumulant [@moments]. (Here $\rho$ is the density of active sites.) That is, defining $\lambda_\times (L)$ via the condition $m[\lambda_\times (L);L] = m[\lambda_\times (L);L\!-\!1]$, the $\lambda_\times (L)$ converge to $\lambda_c$ as $L \to \infty$, as follows from a scaling property of the order-parameter probability distribution. Our procedure, therefore, is to calculate $m(\lambda;L)$ for a series of sizes $L$, locate the crossings $\lambda_\times (L)$, and use them to estimate $\lambda_c$.

In one dimension we calculate $m(\lambda;L)$ for rings of size $L=6$ to 11. We treat configurations with only one species as absorbing, as well as, naturally, the configuration devoid of any individuals. To estimate $\lambda_c$ we perform a quadratic fit to $\lambda_\times (L)$ as a function of $L^{-\gamma}$, using $\gamma$ in the range 1-3. (The precise value of $\gamma$ is chosen so as to render the plot of $\lambda(L)$ versus $L^{-\gamma}$ as close to linear as possible.) Similar estimates for $\lambda_c$ are obtained using the Bulirsch-Stoer procedure [@BS]. As is evident in Fig. \[lm1d\], the resulting phase boundary is in good accord with simulation, predicting $\lambda_{c,\mu}$ with an accuracy of 10% or better. 