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Space used by animals increases with increasing body size. Energy requirements alone can explain how population density decreases, but not the steep rate at which home range area increases. We present a general mechanistic model that predicts the frequency of interaction, spatial overlap, and loss of resources to neighbors. Extensive empirical evidence supports the model, demonstrating that spatial constraints on defense cause exclusivity of home range use to decrease with increasing body size. In large mammals, over 90% of available resources may be lost to neighbors. Our model offers a general framework to understand animal space use and sociality.