The science
Escape geometry. A rabbit a distance s from its burrow runs at speed ur; a fox d behind chases at uf. In a straight run the rabbit is safe if s/ur < (s + d)/uf, i.e. the fox must be detected before it is closer than the critical distance d* = s (uf/ur − 1). Graze closer to the burrow and d* shrinks.
Detection. The fox stalks in from d0 at speed us. A head-up rabbit notices it with hazard λ0; head-down it notices nothing. With a fraction v of the time head-up the chance to detect it before d* is
P_escape = 1 - exp( -lambda0 v (d0 - d*) / u_s ).
Optimal vigilance. Intake scales as (1 − v) while the risk of an undetected approach scales as e−cv, c = λ0(d0 − d*)/us. Maximising gain per unit mortality, (1 − v) ecv, gives v* = 1 − 1/c (0 if c ≤ 1). Long burrow distances, fast foxes and low detection ability push c down and make bold feeding the best strategy; a wide-open approach path makes watchfulness pay.
Many eyes. n rabbits that watch independently detect the fox with hazard n v λ0, so each can look up less in a larger group - one reason rabbits and many birds feed in flocks.
Grazing. Bite rate follows a Holling type-II curve a G / (1 + a h G) times the head-down fraction; grass grows logistically, so heavily grazed patches near the burrow recover slowly and rabbits are pulled outward.
Simplifications: 2-D meadow, one fox, straight-line stalk and chase (no zig-zag), constant sprint speeds, constant detection hazard, no vegetation cover or smell. All rates (hazard, speeds, regrowth, bout length) are illustrative round numbers, not measurements for a particular population. The Monte-Carlo detection process alternates head-up and head-down bouts, so its escape probability sits a little below the continuous-hazard formula. Rabbits lost are simply removed and replaced by newcomers; no harm is depicted.
Independent educational model; animals are low-poly illustrations.