Each worm is an active Brownian particle: it crawls at constant speed v while its heading undergoes rotational diffusion, dθ = √(2Drotdt)·N(0,1). This "persistent random walk" is the standard model for animal search movement (used for everything from C. elegans locomotion to bacterial run-and-tumble). It has a closed-form mean-squared-displacement:
MSD(t) = 2(v²/D_rot²)·(D_rot·t − 1 + e^(−D_rot·t))
τ = 1 / D_rot (persistence time)
D = v² / (2·D_rot) (long-time diffusion coeff.)
At short times (t≪τ) motion is ballistic, MSD≈v²t²; at long times it becomes diffusive, MSD≈4Dt. The top-right plot overlays the population-averaged measured MSD against this exact curve — watch it peel away once worms start bouncing off the garden's edges, since the formula assumes unbounded space.
Finding food is modelled with the classic random-search encounter-rate result: a searcher of detection radius r moving at speed v sweeps area at rate 2rv, so against a leaf density ρ it should find food at rate λ = 2rvρ per worm. With N worms and F(t) leaves left in garden area A, that predicts exponential depletion F(t) = F₀·e^(−2Nrv·t/A) — plotted against the actual (stochastic, discrete) leaf count below.
- gu — "garden units", an abstracted length scale so the search-theory formulas stay dimensionally clean.
- Both theory curves assume worms don't interfere with each other or the walls — real trajectories diverge from them as the garden fills up or empties out.