Transition rule complexity in grid-based automata models

Grid-based automata models have been widely applied in simulating ecological process and spatial patterns at all spatial scales. In this paper, we present methods for calculating the effects of number of states, size of the neighborhood, means of tallying neighborhood states, and choice of deterministic or stochastic rules on the complexity and tractability of spatial automata models. We use as examples Conway's Game of Life and models for successional dynamics in a mesquite savanna landscape in south Texas. The number of possible neighborhood state configurations largely determines the complexity of automata models. The number of different configurations in Life, a two-state, deterministic, voting-rule model with an eight-cell Moore neighborhood is 18. A similar model for the seven-state savanna system would have 21,021 different neighborhood configurations. For stochastic models, the number of possible state transitions is the number of neighborhood configurations times the number of possible cell states. A stochastic, unique neighbor model for the savanna system with a Moore neighborhood and seven possible states would have 282,475,249 possible neighborhood-based state transitions. Stochastic models with an eight-cell Moore neighborhood are probably most appropriate for ecological applications. The best options for minimizing the complexity of ecological models are using voting rather than unique neighbor transition rules, reducing the number of possible states, and implementing ecologically-based heuristics to simplify the transition rule table.