  This article is concerned with the following spectral problem: to find a
positive function $\Phi$ $\in$ C 1 ($\Omega$) and $\lambda$ $\in$ R such that
q(x)$\Phi$ (x) + ^ $\Omega$ J(x, y)$\Phi$(y) dy + a(x)$\Phi$(x) +
$\lambda$$\Phi$(x) = 0 for x $\in$ $\Omega$, where $\Omega$ $\subset$ R is a
non-empty domain (open interval), possibly unbounded, J is a positive
continuous kernel, and a and q are continuous coefficients. Such a spectral
problem naturally arises in the study of nonlocal population dynamics models
defined in a space-time varying environment encoding the influence of a climate
change through a spatial shift of the coefficient. In such models, working
directly in a moving frame that matches the spatial shift leads to consider a
problem where the dispersal of the population is modeled by a nonlocal operator
with a drift term. Assuming that the drift q is a positive function, for rather
general assumptions on J and a, we prove the existence of a principal eigenpair
($\lambda$ p , $\Phi$ p) and derive some of its main properties. In particular,
we prove that $\lambda$ p ($\Omega$) = lim R$\rightarrow$+$\infty$ $\lambda$ p
($\Omega$ R), where $\Omega$ R = $\Omega$ $\cap$ (--R, R) and $\lambda$ p
($\Omega$ R) corresponds to the principal eigenvalue of the truncation operator
defined in $\Omega$ R. The proofs especially rely on the derivation of a new
Harnack type inequality for positive solutions of such problems.
