Evolutionary foraging in grids: Intermittent search emerges as an optimal strategy

How search strategies evolve in sparse, depletable landscapes remains a central question in foraging theory. We study this problem with an evolutionary simulation in which agents forage on a two-dimensional toroidal lattice containing non-renewable resources distributed either uniformly or as Lévy dust. Each agent carries a heritable genome encoding step lengths, velocities, and turning angles, and selection acts on a fitness function, derived from first principles, combining energetic gain, movement cost, and coverage efficiency. By allowing movement traits to evolve without imposing a prescribed power-law step-length distribution, we test whether selection recovers a strict Lévy-like random walk, similar to the spatial distribution of resources, or instead favors an alternative search mechanism. Strikingly, our results indicate that, in the finite depletion-driven landscapes considered here, evolved search is more consistent with intermittent dynamics than with strict scale-free Lévy motion. To characterize the effective dynamics of the evolved trajectories, we take the average coefficient of determination when fitting second- and fourth-order displacement moments to intermittent search and Lévy-walk models. While a Lévy-like random walk fits very well with the numerical results from the evolutionary search ($R^2>0.9$), the intermittent search achieves a closer fit, with fitted coefficients of determination ($R^2>0.99$) for all resource distributions. Evolution rapidly reshapes the movement genome toward short displacements while retaining a sparse tail of longer relocations, consistent with local exploitation punctuated by occasional transfer. The framework provides a controlled setting for studying how search rules emerge under resource limitation and may inform resource-constrained exploration in autonomous systems.