The pre-eminent model of flight initiation distance assumes that the function relating predation risk to distance between predator and prey is constant. However, the risk–distance function can change dramatically during approaches by predators. Changes in predator behavior during approach and in availability of benefits (e.g. food or potential mates) may alter risks and/or costs during encounters. Thus, prey should be able to respond appropriately to changes in cues to risk, such as predator approach speed. Under the assumption that prey assess risk in real time, it was predicted that flight initiation distance (distance between predator and prey when escape begins) decreases when approach speed increases and increases when approach speed decreases during an encounter. Effects of single, abrupt changes from slower to faster approach or the reverse were studied in a lizard, Anolis lineatopus. Flight initiation distances were determined solely by final approach speed, being nearly identical for: (1) continuously fast approaches and approaches initially at the slower and finally at the faster speed and (2) for continuously slower approaches and approaches initially at faster and finally at slower speed. Escape should be adjusted to match changes in risk and cost caused by changes in predator behavior, ability to escape, and costs of escape as attacks unfold. A recent model by Broom and Ruxton [Behavioural Ecology (2004) vol. 16, pp. 534—540] predicts that cryptic prey should stay motionless until detected, then flee immediately. Our results suggest that current escape models can be applied to prey escape strategies when cues to risk change, by assuming that prey base decisions on the current relationship between risk and distance. Empirical studies are needed to test predictions concerning continuous risk assessment.