Analysis of the Lévy Flight Foraging Hypothesis in \(\mathbb{R}^{n}\) and Unreliability of the Most Rewarding Strategies

The Lévy Flight Foraging Hypothesis in Bounded Regions

“…One may wonder whether the unreliability of the most rewarding strategies and the corresponding high-risk/high-reward searching mode are specific of the situation considered in this memoir, i.e., of a forager confined in a bounded region and a nearby prey. This is not the case, in fact in the paper [13] we will show that the same pattern persists, for instance, for a predator diffusing in the whole space and also for a prey located arbitrarily far from the predator.…”

“…The situation in which the biological population is not confined into a bounded region of space and can travel through the whole of R n is technically different and has been treated in the papers [12,13].…”

“…Interestingly, in our setting, the situation in which the most rewarding strategy is arbitrarily close to a complete failure of the searching pattern is related to Lévy distributions with a very low exponent and a very fat tail. This pattern is known to be related to foraging modes of "ambush" type (see [13,16]). The literature has also collected experimental evidence of some species, such as anglers and blonde skates, which do follow diffusive paths with very low Lévy exponent: remarkably, a correlation has been found between this type of diffusion and the high content of energy of the targets (see [16]).…”

“…The case that will be addressed in [13] is technically different from the one here, since the spectral analysis cannot be performed in unbounded domains and we will have to rely on singular integral calculations instead.…”

The Lévy flight foraging hypothesis: comparison between stationary distributions and anomalous diffusion *

“…In this context, we have recently introduced and discussed several 'efficiency functionals' related to animal foraging and motivated by the Lévy flight foraging hypothesis, see [DGV22,DGV23,DGVa,DGVb,DGVc]. In a nutshell, the gist of these functionals is to consider the distribution of foragers and preys as probability densities and evaluate the outcome of the search from the random encounters of foragers and preys, as encoded by the product of the corresponding distributions.…”