Letter: A comment on Vine's predator-prey visual detection model.

Vine (1971) examines different geometrical spacings of prey in order to determine their risk of visual detection and pursuit by a predator. In his first model, Vine considers a situation where there are n prey (es), each of body length I and all at a distance r from some predator P. This predator can detect a prey (occupying Z/r radians) if that prey is standing completely within 8, radians in front of him (f7,/2 on either side of Ps frontal axis). The predator begins searching for the prey at some angle o! (of initial fixation. This angle is assumed to be uniformly distributed over 2n radians.) P sweeps at a constant rate CJ~ radians/unit time. If there were only one prey and if a = (&-l/r)/2 relative to dead center on the prey, then P will be able to just detect the prey with detection time = 0. If 6: is just slightly larger, then P will have to sweep before finding the prey. The longest detection time is when P just misses the prey (at a = 2n -(0,-I/r)/2) and then sweeps in the direction away from the prey. Vine is attempting to show that if we consider the case where there are n prey, the time before any one of the prey is detected will be maximal if they all group together in one nose-to-tail string. In order to show this, Vine determines the mean detection time for the two most extreme spacing configurations. For the case of n nose-to-tail prey (all on the circle of radius r from P) he determines T,, the mean detection time pine, 1971, p. 411 equation (1 b)] to be :