A forager with limited survival time S searches for food on a lattice. At each food site, the forager consumes only partially — eating until its energy reaches a threshold k, then moving on. The fraction consumed depends on how depleted the forager is when it arrives: a starving forager eats more than a well-fed one.
The forager's lifetime increases with the threshold-to-survival ratio k/S. Eating more at each site means fewer sites needed, fewer steps between meals, less energy spent traveling. But the increase is not uniform. Below a critical threshold k ~ √S, lifetime increases rapidly with k/S. Above k, the improvement saturates — additional consumption per site yields diminishing returns.
The transition at k* separates two foraging regimes. Below the threshold, the forager is genuinely energy-limited: most of its time is spent searching, and any increase in per-site intake dramatically extends life. Above the threshold, the forager is exploration-limited: it has enough energy from each site but spends time returning to previously depleted sites, and eating more at each visit doesn't help because the constraint is spatial, not energetic.
The lifetime follows a power law τ ~ S^β, with β varying continuously from 4/3 at k/S = 0 (no partial consumption, pure starvation dynamics) to approximately 1.84 at large k/S (efficient consumption, survival limited by exploration). The exponent itself is a function of the strategy. The forager's behavioral parameter — how much to eat before moving on — doesn't just shift a curve; it changes the scaling law that governs the curve. Strategy controls the universality class of survival.