Durrett and Levin (1994) proposed that the behavior of stochastic spatial models could be determined from the properties of the mean field ODE, i.e., the equations for the densities which results from pretending that adjacent sites are independent.
Case 1. When there is a single attracting fixed point with all components positive, there will be coexistence in the spatial model as well.
Case 2. There are two locally attracting fixed points in the ODE, so the limiting behavior depends on the initial densities. In the spatial model, there is one stronger equilibrium that is the winner starting from generic initial conditions. To determine the stronger equilibrium, one starts with one half plane in each equilibrium and watches the direction of movement of the front.
Case 3. Periodic orbits in the ODE. In the spatial model densities fluctuate wildly on small length scales, oscillate smoothly on moderate length scales, and after an initial transient are almost constant on large scales. That is, there is an equilibrium state with an interesting spatial structure. For attempts at defining the moderate length scale precisely see Rand and Wilson (1995) and Keeling et al (1996).
These principles are a heuristic guide, but there is a growing list of examples where the conclusions have been demonstrated by simulation or proved mathematically.
Durrett, R. and Levin, S. (1994) The importance of being discrete (and spatial). Theoret. Pop. Biol. 46, 363-394
Rand, D.A. and Wilson, H.B. (1995) Using spatio-temporal chaos and intermediate scale determinism in artificial ecologies to quantify spatially-extended systems. Proc. Roy. Soc. London. 259, 111-117
Keeling, M.J., Mezic, I., Hendry, R.J., McGlade, J., and Rand, D.A. (1996) Characteristic length scales of spatial models in ecology. Preprint
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