  We study the positive principal eigenvalue of a weighted problem associated
with the Neumann spectral fractional Laplacian. This analysis is related to the
investigation of the survival threshold in population dynamics. Our main result
concerns the optimization of such threshold with respect to the fractional
order $s\in(0,1]$, the case $s=1$ corresponding to the standard Neumann
Laplacian: when the habitat is not too fragmented, the principal positive
eigenvalue can not have local minima for $0<s<1$. As a consequence, the best
strategy for survival is either following the diffusion with $s=1$ (i.e.
Brownian diffusion), or with the lowest possible $s$ (i.e. diffusion allowing
long jumps), depending on the size of the domain. In addition, we show that
analogous results hold for the standard fractional Laplacian in $\mathbb{R}^N$,
in periodic environments.

Analysis of PDEs