Here the possible states are 0 = grass, 1 = bushes, 2 = trees. 0's are thought of as vacant sites. Types 1 and 2 behave like contact processes subject to the rule that 2's can give birth onto sites occupied by 1's but not vice versa. In formulating the dynamics (and naming the model) we are thinking of the various types as species that are part of a successional sequence. For more on this and a multi-species generalization see Tilman (1994).
1's die at a constant rate d[1] and are born at sites in state 0 at rate b[1] times the fraction of neighbors in state 1.
2's die at a constant rate d[2] and are born at sites in state 0 or 1 at rate b[2] times the fraction of neighbors in state 2.
2's don't feel the presence of 1's, so they are a contact process and will survive if d[2]/b[2] < delta_c. The main question then is: when can the 1's survive in the space that is left to them?
To investigate this question Durrett and Swindle (1991) considered what happens when the dispersal distances are large, e.g., two sites are called neighbors if they differ by at most r is any coordinate. As in the case of the long range contact process, the motivation is that in this case the densities will behave like solutions to the mean field ODE:
The equilibirum density of 2's will be v[2] = (b[2]-d[2])/b[2]. Inserting this into the first equation and solving one finds there is an equilibrium with v[1] > 0 if
As written, this condition was derived by asking the question: Can the 1's invade the 2's when they are in equilibrium? That is, u[1] will increase if when it is small enough.
The next two results say that when the range r is large enough the spatial model behaves like the ODE.
Durrett and Swindle (1991). If (A) holds then coexistence occurs for large range.
Durrett and Schinzai (1993). Suppose < in (A). If the range is large 1's die out. That is, if there are infinitely many 2's in the initial configuration then P( state_t[x] = 1 ) tends to 0 as t approaches infinity.
s3 Exercise. Set b[2] = 2.0, d[1] = d[2] = 1.0. The results above imply that when the range is large we will have survival if and only if b[1] > 4. Start with all sites full and try various values of b[1] to estimate the critical value when the range = 4.
Other related work. The results in Durrett and Schinazi (1993) also apply to the Crawley and May's (1987) model of the competition between annuals (1's) and perennials (2's). In this case the perennials are a nearest neighbor contact process but annuals have a long dispersal distance.
Crawley, M.J. and R.M. May (1987) Population dynamics and plant community structure: competition between annuals and perennials. J. Theor. Biol. 125, 475-489
Durrett, R. and Swindle, G. (1991) Are there bushes in a forest? Stoch. Proc. Appl. 37, 19-31
Durrett, R. and Schinazi, R. (1993) Asymptotic critical value for a competition model. Ann. Applied. Prob. 3, 1047-1066 Tilman, D. (1994) Competition and bio-diversity in spatially structured habits. Ecology. 75, 2-16