Lévy flight random searches in biological phenomena

There has been growing interest in the study of Lévy flights observed in the movements of biological organisms performing random walks while searching for other organisms. Here, we approach the problem of what is the best statistical strategy for optimizing the encounter rate between "searcher" and "target" organisms - either of the same or of different species - in terms of a limiting generalized searcher-target model (e.g., predator-prey, mating partner, pollinator-flower). In this context, we discuss known results showing that for fixed targets an inverse square density distribution of step lengths can optimize the encounter rate. For moving targets, we review how the encounter rate depends on whether organisms move in Lévy or Brownian random walks. We discuss recent findings indicating that Lévy walks confer a significant advantage for increasing encounter rates only when the searcher is larger or moves rapidly relative to the target, and when the target density is low.