Visual experience involves not only physical features such as color and shape, but also higher-level properties such as animacy and goal-directedness. Perceiving animacy is an inherently dynamic experience, in part because agents' goal-directed behavior may be frequently in flux--unlike many of their physical properties. How does the visual system maintain and update representations of agents' animate and goal-directed behavior over time and motion? The present study explored this question in the context of a particularly salient form of perceived animacy: "chasing", in which one shape (the "wolf") pursues another shape (the "sheep"). Here the participants themselves controlled the movement of the sheep, and the perception of chasing was assessed in terms of their ability to avoid being caught by the wolf--which looked identical to many moving distractors, and so could be identified only by its motion. The wolf's pursuit was frequently interrupted by periods in which it was static, jiggling in place, or moving randomly (amidst distractors that behaved similarly). Only the latter condition greatly impaired the detection of chasing--and only when the random motion was grouped into temporally extended chunks. These results reveal (1) how the detection of chasing is determined by the character and temporal grouping (rather than just the brute amount) of "pursuit" over time; and (2) how these temporal dynamics can lead the visual system to either construct or actively reject interpretations of chasing. (Contains 15 figures, 6 tables, and 4 footnotes.)