Lagniappe #9
Simulation of the Nonlinear Voter Model
adapted from Local Frequency Dependence and Global Coexistence
© J. Molofsky, R. Durrett, S. Levin, J. Dushoff and D. Griffeath, 1997
 Nearly three years ago, in our December 12, 1994 recipe, we described an interesting instance of phase separation within a simple twoparameter class of probabilistic cellular automata known as the Nonlinear Voter Model. What follows is an account of our extensive simulation of those systems over the entire range of parameter values, in order to obtain a good approximation to the phase portrait. These findings will be summarized in a forthcoming article by the abovementioned authors.
Let us investigate the various types of system behavior which can arise in the Nonlinear Voter Model for different choices of the parameters p_{1} and p_{2}. In light of meanfield analysis, which ignores spatial interactions, we should anticipate a similarly complex phase portrait, but the extent of commonality is hard to predict. Unfortunately, except in the linear voter model case (.2,.4), the exact symmetry of our automata places them beyond the reach of currently available techniques for rigorous mathematical analysis. Thus we resort to Monte Carlo simulation in order to compare and contrast the phase portrait of the spatial model with that of its nonspatial counterpart.
 The meanfield portrait suggests that evidence of ergodicity, clustering, and phase separation may be most easily discernible with parameters on the bottom edge, { (p_{1},0) ; 0 < p_{1} < 1 ) }. In order to test for qualitatively different spatial behaviors, we use our WinCA modeling software to check how readily each species is able to invade the other's terrain. Our experiments take place on a 640 by 480 array, with type 0 (white) initially on the left half, type 1 (black) on the right half, and with a mixed boundary condition which wraps the top edge to the bottom but disables interaction between the left and right edges. This setup effectively emulates largescale interface dynamics of the infinite system. We also analyze more conventional simulations, starting from random initial configurations with a given density in (0,1), and with wraparound at each edge. As it turns out, we encounter three distinct scenarios by varying p_{1}, with p_{2} = 0.
 If p_{1} = .35, or for any larger value of the parameter, each species invades the other's territory at a linear rate, and between the two wave fronts the system rapidly reaches an equilibrium with density .5. Fig. 1 shows this process after 1000 updates.
