Direct comparison is virtually impossible, because even the local description is 64 dimensional. Births occur onto neighboring empty sites, and predators and prey need only be in the same neighborhood to affect each other's birth and death rates. All three models are fundamentally similar in their assumptions and are based on the multi-state equations above. The death expressions from these equations remain exactly the same, with the exception that numbers of predators and prey are counted within the neighborhood. The birth probability, however, now represents the probability of a birth onto an empty cell given that the empty cell has P 1 and P 2 neighboring prey of types 1 and 2 respectively, and P d neighboring predators.
Transport involves random movement from an occupied to an unoccupied neighboring cell.
The probabilities of sites transitioning from one state to another for the presence-absence model are described in Figure 2. Instead of P x representing the percent of patches in the environment occupied by type x , it represents the percent of patches in a given neighborhood that are of that type. Additionally, P' x represents the probability of any given cell being in any one of the three given states in the next time step. For the global dispersal model, the neighborhood is the whole grid, and transport is ignored since any movement would be irrelevant. I divide discussion into two main cases, as distinguished by the dynamics of the mean field approximation which has several possible equilibria.