Best dispersal strategies in spatially heterogeneous environments: optimization of the principal eigenvalue for indefinite fractional Neumann problems

Abstract: 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.