  Motivated by experimental studies on the anomalous diffusion of biological
populations, we introduce a nonlocal differential operator which can be
interpreted as the spectral square root of the Laplacian in bounded domains
with Neumann homogeneous boundary conditions. Moreover, we study related linear
and nonlinear problems exploiting a local realization of such operator as
performed in [X. Cabre' and J. Tan. Positive solutions of nonlinear problems
involving the square root of the Laplacian. Adv. Math. 2010] for Dirichlet
homogeneous data. In particular we tackle a class of nonautonomous
nonlinearities of logistic type, proving some existence and uniqueness results
for positive solutions by means of variational methods and bifurcation theory.
