Should co-operative groups be more vigilant than selfish groups?

We develop four simple models of group vigilance that illustrate the conditions under which co-operative groups should be more or less vigilant than selfish groups. In the first model, prey individuals that directly detect the predator have a lower chance of being captured than those that were not vigilant at the time of the attack; but the predator's choice of a specific individual is not influenced by the prey's behavior. In this case, the expected level of vigilance in a co-operative group always equals or exceeds that in a selfish group. In the second model, the predator always attacks an individual that was not vigilant at the time of the attack. Co-operative vigilance can now be lower than selfish vigilance, providing that being warned by a vigilant flockmate has a sufficiently small effect on the chances of escape by a nonvigilant group member. The third model is a generalization of the first two models for the case of small groups. It can be used to consider all situations in which the predator bases its selection on the prey's behavior at the time of its attack. It reveals a variety of conditions in which selfish groups are expected to be more vigilant than co-operative groups. In the fourth model, the predator assesses the vigilance levels of each prey prior to its attack and attacks the individual that it judges to be least vigilant. If the predator is sufficiently accurate in assessing the least vigilant prey, selfish groups are always more vigilant than co-operative groups. In this case, there are circumstances where a selfish population will be expected to undergo endless cycles in its level of vigilance.