An okapi strips leaves from shrubs along a forest trail. Each shrub gets harder to eat as it empties, and the next one is a walk away. When should it move on?
The curve is the energy eaten in one shrub. The dashed line starts at the walking time (left of zero) and just touches the curve at the best leaving time t*. Its slope is the best long-run rate.
Simulates 400 shrubs per threshold (shrub richness varies randomly) for threshold multipliers 0.2 to 3 and compares the measured mean rate with the formula. The prediction is a peak at multiplier 1.
The science
Patch depletion. A shrub holds digestible energy Ge (kJ). Browsing removes it at a rate proportional to what is left: intake rate r(t) = r0 e-bt with b = r0/Ge, so the energy eaten after time t is g(t) = Ge(1 - e-bt).
Marginal value theorem (Charnov 1976). Between shrubs the animal walks for τ = d / v and pays cw = ew d kJ. Leaving each shrub when its intake rate has fallen to γ gives a long-run net rate R = (E[gain] - cw) / (E[residence] + τ). With the exponential gain this becomes R(u) = (Ge(1-u) - cw) / ((Ge/r0) ln(1/u) + τ) where u = γ/r0. The best rule is to leave when the current intake rate equals the average rate you can achieve, γ* = R(γ*). Longer walks give a lower γ*, so the animal stays longer on each shrub. Poor shrubs are left earlier in time, but at the same intake rate.
Tannins. Condensed tannins bind leaf protein. The model lowers digestible energy by fD = 1 - 0.7 T/(T+0.4) and lowers the bite rate by 1 - 0.5 T, T = 0 to 1 (a defence index, not a measured concentration).
Energy budget. Walking costs 0.7 kJ per metre (cost of transport about 2.8 J per kg per m for 250 kg). A daily need of 30 MJ is about 1.6 times the Kleiber basal rate 3.4 M0.75 W. These are typical orders of magnitude, not measurements of okapi.
Honest simplifications. One animal, no regrowth, no predators, no water or minerals, no learning, no memory of poor shrubs, no limit on rumen fill or toxin load, constant walking speed, and a perfectly known average rate (real animals must estimate it). Shrub richness is log-normal with fixed spread 0.5. All rate constants are illustrative; the okapi (Okapia johnstoni, a shy forest giraffid of the Congo basin) has not been studied in enough detail to fit them. The drawing is a time-lapse, not real walking speed.
Models built with Blender (okapi, trees, shrubs, ferns, forest floor) and rendered with Three.js.