Serena Dipierro, Giovanni Giacomin, and Enrico Valdinoci
The L\'evy flight foraging hypothesis in bounded regions:
subordinate Brownian motions and high-risk/high-gain strategies
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ABSTRACT. We investigate the problem of the L\'evy 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 $s \in (0, 1)$ 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 L\'evy 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 high-risk/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.