Analysis of the Lévy Flight Foraging Hypothesis in R^n and Unreliability of the Most Rewarding Strategies

We analyze the searching strategies of a forager diffusing in the whole space via an equation of fractional type. Specifically, the diffusion of the forager is regulated by a Lévy flight whose exponent can be chosen in order to optimize a suitable foraging efficiency functional. Here, the dimension of the space is arbitrary. On the one hand, we show that the exponent corresponding to the limit case of heavy-tailed Lévy flights is a pessimizer for the efficiency functional. On the other hand, we prove that, in situations of biological interest, one finds the most rewarding strategies arbitrarily close to. The combination of these results gives that the most rewarding searching option may turn out to be unfeasible, or at least unreliable, in practice, since small perturbations of the optimal searching exponent lead to pessimal patterns. The cases analyzed specifically are those of a target located in the proximity of the forager and that of sparse prey modeled by a target infinitely far from the initial position of the seeker. The efficiency functionals taken into account are either of pointwise type (in which the predator and the prey are modeled by moving points) or of set-dependent type (in which the predator and the prey correspond to regions of space with uniform density, thus modeling also the case of a sight range of the biological individuals involved). To implement our analysis, we also provide a number of structural results about finiteness, continuity, and asymptotic behaviors of the efficiency functionals. It is suggestive to relate the adoption of the most rewarding searching pattern close to pessimizers to a “high-risk/high-gain” strategy, in which the forager aims at high-energy content prey to mitigate the risk of failure. This setting is also connected to foraging modes of “ambush” type.