The Why of Waiting: How mathematical Best-Choice Models demonstrate optimality of a Refractory Period in Habitat Selection

When brush mice, fruit flies, and other animals disperse from their natal site, they are immediately tasked with selecting new habitat, and must do so in such a way as to optimize their chances of surviving and breeding. Habitat selection connects the fields of behavioral ecology and landscape ecology by describing the role the physical quality of habitat plays in the selection process. Interestingly, observations indicate a strategy that occurs with a certain prescribed statistical regularity. It has been demonstrated (Stamps, Davis, Blozis, Boundy-Mills, Anim. Behav., 2007) that brush mice and fruit flies employ a refractory period: a period wherein a disperser, after leaving its natal site, will not accept highly-preferred natural habitats. Assuming this behavior has adaptive benefit, the apparent optimality of this strategy is mirrored in mathematical models of Stochastic Optimization. In one such model, the Classical Best Choice Problem, a selector views some permutation of the numbers {1, ..., n} one-by-one, seeing only their relative ranks and then either selecting that element or discarding it. The goal is to choose the ``n" element. The optimal strategy is to wait for the ⌈ n/e ⌉ ^th element and then pick an element if it is better than all those already seen; this might demonstrate why refractory periods have adaptive benefit. We present three extensions to the Best Choice Problem: a partial ordering on the set of elements (Kubicki & Morayne, SIAM J. Discrete Math., 2005), a new goal of minimizing the expected rank (Chow, Moriguti, Robbins, Samuels, Israel J. Math., 1964), and a general utility function (Gusein-Zade, Theory of Prob. and Applications, 1966), allowing the top r sites to be equally desirable. These extensions relate to ecological phenomena not represented by the Classical Problem. In each, we discuss the effect on the duration or existence of the Refractory Period.