  This work examines the limits of the principal spectrum point, $\lambda_p$,
of a nonlocal dispersal cooperative system with respect to the dispersal rates.
In particular, we provide precise information on the sign of $\lambda_p$ as one
of the dispersal rates is : (i) small while the other dispersal rate is
arbitrary, and (ii) large while the other is either also large or fixed. We
then apply our results to study the effects of dispersal rates on a two-stage
structured nonlocal dispersal population model whose linearized system at the
trivial solution results in a nonlocal dispersal cooperative system. The
asymptotic profiles of the steady-state solutions with respect to the dispersal
rates of the two-stage nonlocal dispersal population model are also obtained.
Some biological interpretations of our results are discussed.
