Many-eyes effect. Each second an animal has its head up with probability b. If the threat is inside the visibility radius d_vis, any head-up animal notices it. With N independent animals the group notices within a second with probability p = 1 - (1 - b)^N.
Detection distance. The threat closes at speed u. After crossing d_vis the number of whole seconds K before the first detection is geometric with mean E[K] = (1-b)^N / (1 - (1-b)^N), so the expected detection distance is d_vis - u E[K] (point-herd approximation; the checks panel compares it with a Monte-Carlo run).
Fixed alarm rate. Keeping p constant gives per-animal vigilance b = 1 - (1-p)^(1/N), which falls steeply as the herd grows, and intake rises in proportion to 1 - b.
Visibility. d_vis = 140 m x (1 - 0.6 g) for grass height g between 0 and 1 (illustrative). Real animals also scan in bouts, copy neighbours and sometimes cluster for dilution; those effects are ignored, and so are predator learning and kills: the threat only stalks, and the herd flees once an animal has seen it. All constants are teaching values.