TRANSIENT DYNAMICS AND QUASISTATIONARY EQUILIBRIA OF CONTINUOUS-TIME LINEAR STOCHASTIC CELLULAR AUTOMATA VOTER MODELS WITH MULTISCALE NEIGHBORHOODS

This paper studies asynchronously-updated linear voter cellular automata models, with local and global interactions. In the locally-interacting model, sites copy the states of randomly-chosen sites in the local neighborhood; in the globally-interacting model, sites copy the states of sites chosen at random from the entire lattice. A multiscale model is also considered, mixing the two types of interactions. Such models can be used to model simple genetic drift, or the spread of opinions or ideas. The pair approximation moment-closure method is used to develop systems of differential equations describing the dynamics of the models with k states per site; stochastic spatially explicit simulations are also used. In simulations on large lattices, state frequencies among sites remain relatively constant at their initial values, while autocorrelations between adjacent sites move toward quasiequilibrium distributions and then remain constant with minor stochastic fluctuations. Full analytical solutions for the transient dynamics of the local autocorrelation (clustering) among adjacent sites are obtained with k = 2 states, and compared with simulations. Both show that increasing global interactions decreases spatial autocorrelation, and that even in the absence of long-distance interaction, there will still be mixing among the states. That is, the lattice will not converge to a quasiequilibrium configuration where regions in different states are maximally isolated from each other (completely segregated), but instead sites will have a strictly positive probability of being adjacent to sites in different states.