Background/Question/Methods Savanna ecosystems are widespread, ecologically and economically important, and despite considerable research effort, still not well understood. Research on savannas has focused intensively on elucidating the factors that affect tree-grass coexistence, tree spatial patterning, and bush encroachment. Among the processes so identified are tree dispersal limitation, tree recruitment limitation, fire, and grazing. Models have been used extensively to help understand savanna dynamics, but have tended to be either spatially implicit analytical models focusing on one or two factors, or spatially explicit simulation models including these key ingredients among many others. Though these efforts have improved our understanding of savannas, no consensus on the relative importance of and interactions among the key processes has yet emerged. We argue that this lack of progress is partly due to savanna models being either too simple and neglecting key interactions or too complex and burying these interactions among many other less important ones. Here, we describe a cellular-automata-based approach that is spatially explicit, incorporates simplified descriptions of the above-mentioned key processes, and is tractable by equation-based analytical techniques. We then derive mean-field and pair approximations of the full model and use these approximations to help understand the qualitative behavior of the model and identify which processes and interactions are most important. Results/Conclusions We demonstrate that trees and grasses coexist over a wide range of conditions, and that a moderately negative effect of nearby established trees on tree seedling recruitment success (hereafter local recruitment limitation) is sufficient to keep tree abundance relatively low. The spatial structure of the tree population is determined by the relative spatial scales and strengths of local recruitment limitation and dispersal limitation. Fire has only limited potential to control tree population size when acting alone, but can strongly suppress tree population growth when coupled with a small degree of local recruitment limitation. Under these conditions, intensive grazing can dramatically alter the tree-grass balance and lead to bush encroachment. We conclude by comparing our combined results to those obtained separately from a range of other modeling approaches.