Lévy flights, optimal foraging strategies, and foragers with a finite lifespan

In some recent work, we have introduced some efficiency functionals to account for optimal dispersal strategies of predators in search of food. The optimization parameter in this framework is given by the L\'evy exponent of the dispersal of the predators. In this paper, we apply our model to the case of foragers with finite lifetime (i.e., foragers which need to eat a certain amount of food in a given time, otherwise they die). Specifically, we consider the case in which the initial distribution of the forager coincides with a stationary distribution of the targets and we determine the optimal L\'evy exponent for the associated efficiency functional. Namely, we show that if the Fourier transform of the prey distribution is supported in a sufficiently small ball, then the optimizer is given by a Gaussian dispersal, and if instead the Fourier transform of the prey distribution is supported in the complement of a suitable ball, then the ballistic diffusion provides an optimizer (precise conditions for the uniqueness of these optimizers are also given).