Search in patchy media: Exploitation-exploration tradeoff.

How to best exploit patchy resources? We introduce a minimal exploitation-migration model that incorporates the coupling between a searcher's trajectory, modeled by a random walk, and ensuing depletion of the environment by the searcher's consumption of resources. The searcher also migrates to a new patch when it takes S consecutive steps without finding resources. We compute the distribution of consumed resources F_{t} at time t for this non-Markovian searcher and show that consumption is maximized by exploring multiple patches. In one dimension, we derive the optimal strategy to maximize F_{t}. This strategy is robust with respect to the distribution of resources within patches and the criterion for leaving the current patch. We also show that F_{t} has an optimum in the ecologically relevant case of two-dimensional patchy environments.