Art de Vany picked up on the Nature paper describing predator foraging behavior in the wild. I was discussion exactly the same problem with Brent Pottenger. Turns out predators use Levy-walks (random walks under a Levy distribution, a.k.a Levy Flights) while foraging. The optimality of levy-walks for predators may hinge on the prey population itself being levy-distributed. It would be interesting to see what happens if the constraint on the prey population is removed.
My latest hypothesis on this is that amongst all stable distributions, the levy-distribution is entropy maximizing under path-length constraints. That is, given finite resources and a fixed distance that a predator can forage, the levy-walk may be entropy maximizing under the set of all possible random walks within the constraint. This problem may prove intractable from an analytical perspective though. We will see.
Most problems in computation rely on optimizing the time to target given unlimited resources, it would be interesting to study how one would optimize probability of encounter given limited resources (battery power, processing time) and completely unknown target distribution.
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