symbiotic contact process, using simulation, cluster approximations, and exact (numerical) quasistationary distributions of small systems. We study the effect of asymmetric creation rates and of diffusion of individuals. Exact quasistationary distributions and cluster approximations provide fair predictions for the phase boundary in the symmetric case. In simulations, the phase transition is always found to be continuous in one dimension, but in two dimensions we observe a discontinuous phase transition when symbiosis is strong ($\mu\to 0$), in the presence of moderate diffusion. For $D \to \infty$ the transition is again continuous.

Although the model studied here is much too simple to apply to real ecosystems, our results raise the possibility of catastrophic (discontinuous) collapse of strongly symbiotic interspecies alliances under increasingly adverse conditions, even if the change is gradual. Possible extensions of this work include precise determination of the tricritical line for the diffusive process, as well as the design of more precise theoretical approaches for two-dimensional problems. The latter task assumes even greater significance when one observes that despite the simplicity of the model, the full parameter space, including distinct reproduction, death, and diffusion rates for each species, is far too vast to be mapped out via simulation alone. Finally, 