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 $\mathcal{S}$ 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.

• Received 16 September 2016

DOI:https://doi.org/10.1103/PhysRevE.95.012157