The Lévy Flight Foraging Hypothesis in Bounded Regions

We investigate the problem of the Lévy flight foraging hypothesis in an ecological niche described by a bounded region of space, with either absorbing or reflecting boundary conditions. To this end, we consider a forager diffusing according to a fractional heat equation in a bounded domain and we define several efficiency functionals whose optimality is discussed in relation to the fractional exponent s ∈ (0, 1) of the diffusive equation. Such equation is taken to be the spectral fractional heat equation (with Dirichlet or Neumann boundary conditions). We analyze the biological scenarios in which a target is close to the forager or far from it. In particular, for all the efficiency functionals considered here, we show that if the target is close enough to the forager, then the most rewarding search strategy will be in a small neighborhood of s = 0. Interestingly, we show that s = 0 is a global pessimizer for some of the efficiency functionals. From this, together with the aforementioned optimality results, we deduce that the most rewarding strategy can be unsafe or unreliable in practice, given its proximity with the pessimizing exponent, thus the forager may opt for a less performant, but safer, hunting method. However, the biological literature has already collected several pieces of evidence of foragers diffusing with very low Lévy exponents, often in relation with a high energetic content of the prey. It is thereby suggestive to relate these patterns, which are induced by distributions with a very fat tail, with a highrisk/high-gain strategy, in which the forager adopts a potentially very profitable, but also potentially completely unrewarding, strategy due to the high value of the possible outcome. Preamble On the one hand, many popular adages share the idea that to achieve a prominent goal one has to take risks (e.g., “no gain without pain”, “nothing ventured, nothing gained”, “no guts, no glory”, just to name a few proverbs). On many occasions, the ambition to a high reward may lead individuals to face potential dangers, and in some situations there is a full master plan centered around a high-risk/high-gain plan: for instance, the blueprint of the European Research Council is to fund high-risk/high-reward research, in which severe conceptual challenges (which, by definition, are prone to scientific failure) are accepted downsides for a research project to be truly successful and impactful. On the other hand, there is nowadays a great interest in the investigation of optimal searching strategies, e.g. in the study of animal behavior, and the research on this topic has necessarily to be somewhat controversial, given the complexity of the phenomenon into consideration. Our view on this point is that the difficulty of addressing the topic of optimal searching is not only due to the enormous amount of parameters which should be accounted for (such as predators and prey distributions, previous knowledge of the territory, interactions with the environment, social factors, different reactions to adverse circumstances, competition phenomena, cooperative behaviors,