The contingencies of group size and vigilance

Background: Predation risk declines non-linearly with one's own vigilance and the vigilance of others in the group (the 'many-eyes' effect). Furthermore, as group size increases, the individual's risk of predation may decline through dilution with more potential victims, but may increase if larger groups attract more predators. These are known, respectively, as the dilution effect and the attraction effect. Assumptions: Feeding animals use vigilance to trade-off food and safety. Net feeding rate declines linearly with vigilance. Question: How do the many-eyes, dilution, and attraction effects interact to influence the relationship between group size and vigilance behaviour? Mathematical methods: We use game theory and the fitness-generating function to determine the ESS level of vigilance of an individual within a group. Predictions: Vigilance decreases with group size as a consequence of the many-eyes and dilution effects but increases with group size as a consequence of the attraction effect, when they act independent of each other. Their synergetic effects on vigilance depend upon the relative strengths of each and their interactions. Regardless, the influence of other factors on vigilance - such as encounter rate with predators, predator lethality, marginal value of energy, and value of vigilance - decline with group size.