Giant Ground Sloth Foraging Map: Marginal Value Theorem, Kleiber Energetics and Diet Choice

Presets

Live readouts

Graphs

Fat reserve in MJ over the simulated days

Daily intake and expenditure in MJ per day

Net rate against minutes on the current patch: average, marginal and the optimal stay

Allometry: resting power, walking power and gut-limited intake in kW against body mass

Diet choice: initial net rate of each plant type against the leave threshold

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Numerical checks

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How to use

Pick a species and a plant community from the presets, or move the sliders. The sloth wakes at the chosen hour, walks to a tree, feeds, and leaves when the marginal rate falls to the leave threshold. Skip 1 day / 30 days fast-forwards; the speed selector sets how fast the clock runs. Compare the three foraging rules with the button under the graphs, switch off a plant type to see how the diet adapts, and push the patience slider to both extremes. Space pauses, R resets. Green graph lines are the leave threshold; the orange dashed line is the marginal rate on the current plant, the blue line the average net rate, and the vertical line the optimal stay.

What to try: (1) Desert scrub with the Shasta sloth: rearing pays less where trees are low. (2) Cold pine woodland: the sloth burns more on heating and the conifer toxins force it to mix foods. (3) Set the search radius to 20 m and watch intake fall. (4) Patience 0.5 versus 2.2: both lose to the middle. (5) Megatherium in a sparse forest: can 4 t be fed at all?

The science

A giant ground sloth had to eat enough leaves, every day, to pay for a very large body. This page turns that into a small, honest energy model: how much it costs to live, how much a patch of browse yields, and when to stop eating one tree and walk to the next. The same session drives the 2D map and the 3D forest, so both give identical numbers.

1. Body-mass scaling. Resting power follows Kleiber's law, scaled by a metabolic factor f (living sloths run at roughly half the mammal norm; the value for extinct species is an inference, not a measurement):

P_rest = f · 70 M^0.75 kcal/day, P_walk = C_T(M) · M · v, C_T = 10.7 M^−0.316 J kg⁻¹ m⁻¹ (Taylor et al. 1982)

Below the lower critical temperature T_lc = 22 − 2 log₂(M/500) °C the resting power rises by 3 % per degree (an assumed linear heating cost), so a cold glacial woodland is expensive for a hairy animal that cannot hibernate. Fat is the reserve: 4.56 MJ per kg of body mass at most, 39 MJ per kg of fat burned or stored.

2. A patch of browse. Each tree holds a leaf mass R (kg dry matter) that falls with height as a fraction 1 − e^−h/H_s below height h. The sloth reaches h₀ = 0.15 M^0.33 m on four legs and 1.8 h₀ when it rears up on its hind legs and tail. Intake from a patch is a Holling type-II response to the remaining accessible leaf mass:

dR/dt = − r_max R / (R + K), K = 0.25 A, r_max = r₀ (M/1000)^0.75 · (plant intake factor)

A is the accessible leaf mass of a full tree (smaller when not rearing). Energy gained is the dry matter eaten times the digestible energy of that plant (4.2 to 11 MJ per kg for conifer needles to legume leaves; assumed, order of magnitude). Two pools limit the daily total: a hindgut that holds 1.5 % of body mass and empties at the gut-limited intake 0.07 M^0.75 kg/day, and a toxin pool filled by plant secondary compounds and drained at a constant detoxification rate. When either pool is full the sloth rests until it drops to 85 % (gut) or 70 % (toxin).

3. Marginal value theorem. Charnov (1976): a forager that leaves a patch when the instantaneous net rate there falls to the best long-run average rate γ maximises the long-run net rate. With the type-II depletion above the exact leave amount and residence time are

R* = (c + γ) K / (r_max e − (c + γ)), T* = [ (R − R*) + K ln(R/R*) ] / r_max

where e is the (toxin-discounted) energy per kg and c the extra feeding power. The sloth estimates γ online as an exponentially forgotten average of its realised net rate; the "patience" slider multiplies the threshold (below 1 it over-harvests each tree, above 1 it leaves early and pays in travel). Which tree to visit next is the one with the highest expected net rate including the walk, and rearing is chosen when the extra reachable leaves beat the extra cost. A toxin shadow price lowers the value of toxic plants as the toxin pool fills, which makes a mixed diet emerge.

4. Diet breadth. For a menu of plant types with encounter rates λ_i, energy G_i and handling time h_i, the overall rate is E/T = Σλ_iG_i / (1 + Σλ_ih_i); the optimal menu contains the most profitable types (G_i/h_i) down to the first one whose profitability is below the rate of the menu above it (the zero-one rule).

5. How many sloths could live here? Damuth (1981) found that herbivore population density falls with body mass as D = 10^4.23 M^−0.75 per km² (M in grams). The page prints the density and the implied home-range area for the chosen mass.

Numerical checks run in the page. (1) The closed-form optimal stay on a fresh patch against a brute-force scan of the net rate over an RK4-integrated gain curve; (2) the marginal rate at the optimum equals the best average rate; (3) the zero-one diet rule against enumeration of all 32 possible menus (menu and overall rate); (4) the mean nearest-neighbour distance of the generated forest against the Poisson (Clark-Evans) value 0.5/√density; (5) Kleiber's law against the textbook 70 M^0.75 kcal/day in watts; (6) the walking-energy ledger per metre against the Taylor cost of transport; (7) the reserve change equals intake minus every cost line; (8) Damuth's density for a 1000 kg herbivore.

What the fossils say, and what is assumed. Ground sloths (Megatherium, Paramylodon, Nothrotheriops and relatives) were large xenarthran browsers and grazers that disappeared from the Americas at the end of the Pleistocene; the Shasta ground sloth is known from dung deposits (for example Rampart Cave) rich in desert plants such as yucca, Mormon tea and globemallow. Body masses used here (about 0.5, 1.5 and 4 tonnes) are common estimates. Everything else is a model assumption: metabolic factor, gut size, plant biomass per tree, digestible energies, toxin loads, regrowth rate, reach scaling and search radius. The page demonstrates the logic of optimal foraging and energy budgeting, not a reconstruction of a particular animal.

Honest simplifications. A 240 m × 240 m window of forest, one animal, no predators, no social life, no water or minerals, straight-line walking without obstacles, deterministic plant growth with a seasonal cycle, and a constant detoxification rate. Real foragers also learn, sample, and trade risk against reward; none of that is modelled.