The science
Pure pursuit. The chaser always heads at the target. For a target running in a straight line at speed vg and a chaser at constant speed vc > vg, with φ0 the angle between the target's velocity and the initial line of sight (0 = straight away), the time to reach is
T_pure = d0 (vc + vg cos phi0) / (vc^2 - vg^2)
(Bouguer, 1732). Special cases: straight away gives d0/(vc − vg); head-on gives d0/(vc + vg); a perpendicular runner gives d0 vc/(vc² − vg²). The path is a pursuit curve; the chaser is always behind the target's best direction.
Lead (constant-bearing) pursuit. The chaser aims where the gazelle will be if it keeps going straight: a triangle with sides d0, vgt and vct gives
(vc^2 - vg^2) t^2 - 2 d0 vg cos(phi0) t - d0^2 = 0
and the line of sight keeps a constant direction (zero LOS rate), the signature of interception also reported for hunting falcons and flying insects. Lead is never slower than pure pursuit against a straight runner, but it is fooled by a sudden turn, because the predicted meeting point moves.
Turning. At speed v and maximum lateral acceleration a, the tightest circle has radius r = v²/a; the heading can change at most a/v rad/s. Running fast costs agility: a cheetah at 26 m/s and 14 m/s² needs 48 m to turn round while a gazelle at 20 m/s and 22 m/s² needs only 18 m — the reason a slower prey can win by jinking. The page shows the effect directly: lowering the cheetah's lateral acceleration lengthens the chase.
Simplifications: flat ground, point animals in the plane, constant speeds unless "accelerates from rest" is ticked (simple exponential approach to top speed), no fatigue model other than a time budget, perfect vision, no obstacles or herd. Speeds, accelerations and the budget are plausible round numbers for teaching, not measurements of individual animals. Outcome wording is neutral: "reached" means within 1.2 m, "escaped" means the time budget ran out.
Independent educational model; animals are stylised illustrations.