  We consider here a problem of population dynamics modeled on a logistic
equation with both classical and nonlocal diffusion, possibly in combination
with a pollination term.
  The environment considered is a niche with zero-flux, according to a new type
of Neumann condition.
  We discuss the situations that are more favorable for the survival of the
species, in terms of the first positive eigenvalue.
  Quite surprisingly, the eigenvalue analysis for the one dimensional case is
structurally different than the higher dimensional setting, and it sensibly
depends on the nonlocal character of the dispersal.
  The mathematical framework of this problem takes into consideration the
equation $$ -\alpha\Delta u +\beta(-\Delta)^su =(m-\mu u)u+\tau\;J\star u
\qquad{\mbox{in }}\; \Omega,$$ where $m$ can change sign.
  This equation is endowed with a set of Neumann condition that combines the
classical normal derivative prescription and the nonlocal condition introduced
in [S. Dipierro, X. Ros-Oton, E. Valdinoci, Rev. Mat. Iberoam. (2017)].
  We will establish the existence of a minimal solution for this problem and
provide a throughout discussion on whether it is possible to obtain non-trivial
solutions (corresponding to the survival of the population).
  The investigation will rely on a quantitative analysis of the first
eigenvalue of the associated problem and on precise asymptotics for large lower
and upper bounds of the resource.
  In this, we also analyze the role played by the optimization strategy in the
distribution of the resources, showing concrete examples that are unfavorable
for survival, in spite of the large resources that are available in the
environment.
