Significant progress has been made towards understanding the social behaviour of animal groups, but the patch model, a foundation of foraging theory, has received little attention in a social context. The effect of competition on the optimal time to leave a foraging patch was considered as early as the original formulation of the marginal value theorem, but surprisingly, the role of facilitation (where foraging in groups decreases the time to find food in patches), has not been incorporated. Here we adapt the classic patch model to consider how the trade-off between facilitation and competition influences optimal group size. Using simple assumptions about the effect of group size on the food-finding time and the sharing of resources, we find conditions for existence of optima in patch residence time and in group size. When patches are close together (low travel times), larger group sizes are optimal. Groups are predicted to exploit patches differently than individual foragers and the degree of patch depletion at departure depends on the details of the trade-off between competition and facilitation. A variety of currencies and group-size effects are also considered and compared. Using our simple formulation, we also study the effects of social foraging on patch exploitation which to date have received little empirical study.
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