Abstract
Anti-predator scans by animals occur with very irregular timing, so that the initiation of scans resembles a random, Poisson-like, process. At first sight, this seems both dangerous (predators could exploit the long intervals) and wastefull (scans after very short intervals are relatively uninformative). We explored vigilance timing using a new model that allows both predators and prey to vary their behaviour. Given predators that attack at random with respect to prey behaviour, constant inter-scan intervals minimize predation risk. However, if prey scan regularly to minimize their risk from randomly attacking predators, they become more vulnerable to predators that initiate attacks when the inter-scan intervals begin. If, in order to defeat this tactic, prey choose extremely variable inter-scan intervals, they become more vulnerable to predators who wait for long intervals before launching attacks. Only if predators can monitor the variability of inter-scan intervals and either attack immediately (if variability is too low) or wait for long intervals to attack (if variability is too high) does the empirically observed pattern of Poisson-like scanning become the optimal prey strategy.